Swarm, Evolutionary, and Memetic Computing, Part I - SEMCCO 2011

Lecture Notes in Computer Science Commenced Publication in 1973 Founding and Former Series Editors: Gerhard Goos, Juris Hartmanis, and Jan van Leeuwen

Editorial Board David Hutchison Lancaster University, UK Takeo Kanade Carnegie Mellon University, Pittsburgh, PA, USA Josef Kittler University of Surrey, Guildford, UK Jon M. Kleinberg Cornell University, Ithaca, NY, USA Alfred Kobsa University of California, Irvine, CA, USA Friedemann Mattern ETH Zurich, Switzerland John C. Mitchell Stanford University, CA, USA Moni Naor Weizmann Institute of Science, Rehovot, Israel Oscar Nierstrasz University of Bern, Switzerland C. Pandu Rangan Indian Institute of Technology, Madras, India Bernhard Steffen TU Dortmund University, Germany Madhu Sudan Microsoft Research, Cambridge, MA, USA Demetri Terzopoulos University of California, Los Angeles, CA, USA Doug Tygar University of California, Berkeley, CA, USA Gerhard Weikum Max Planck Institute for Informatics, Saarbruecken, Germany

7076

Bijaya Ketan Panigrahi Ponnuthurai Nagaratnam Suganthan Swagatam Das Suresh Chandra Satapathy (Eds.)

Swarm, Evolutionary, and Memetic Computing Second International Conference, SEMCCO 2011 Visakhapatnam, Andhra Pradesh, India, December 19-21, 2011 Proceedings, Part I

13

Volume Editors Bijaya Ketan Panigrahi IIT Delhi, New Delhi, India E-mail: [email protected] Ponnuthurai Nagaratnam Suganthan Nanyang Technological University, Singapore E-mail: [email protected] Swagatam Das Jadavpur University, Kolkata, India E-mail: [email protected] Suresh Chandra Satapathy ANITS, Visakhapatnam, India E-mail: [email protected]

ISSN 0302-9743 e-ISSN 1611-3349 ISBN 978-3-642-27171-7 e-ISBN 978-3-642-27172-4 DOI 10.1007/978-3-642-27172-4 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011943108 CR Subject Classification (1998): F.1, I.2, J.3, F.2, I.5, I.4 LNCS Sublibrary: SL 1 – Theoretical Computer Science and General Issues

© Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready by author, data conversion by Scientific Publishing Services, Chennai, India Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

This LNCS volume contains the papers presented at the Second Swarm, Evolutionary and Memetic Computing Conference (SEMCCO-2011) held during December 19–21, 2011 at Anil Neerukonda Institute of Technology and Sciences (ANITS), Visakhapatnam, Andhra Pradesh, India. SEMCCO is regarded as one of the prestigious international conference series that aims at bringing together researchers from academia and industry to report and review the latest progress in cutting-edge research on swarm, evolutionary, memetic computing and other novel computing techniques, to explore new application areas, to design new bioinspired algorithms for solving specific hard optimization problems, and finally to create awareness of these domains to a wider audience of practitioners. SEMCCO-2011 received 422 paper submissions in total from 25 countries across the globe. After a rigorous peer-review process involving 1,025 reviews in total, 124 full-length articles were accepted for oral presentation at the conference. This corresponds to an acceptance rate of 25% and is intended for maintaining the high standards of the conference proceedings. The papers included in this LNCS volume cover a wide range of topics in swarm, evolutionary, memetic and other intelligent computing algorithms and their real-world applications in problems selected from diverse domains of science and engineering. The conference featured four distinguished keynote speakers. Carlos. A. Coello Coello’s talk on “Recent Results and Open Problems in Evolutionary Multiobjective Optimization” reviewed some of the research topics on evolutionary multi-objective optimization that are currently attracting a lot of interest (e.g., many-objective optimization, hybridization, indicator-based selection, use of surrogates, etc.) and which represent good opportunities for doing research. Jacek M. Zurada in his talk on “prediction of Secondary Structure of Proteins Using Computational Intelligence and Machine Learning Approaches with Emphasis on Rule Extraction” emphasized the Prediction of protein secondary structures (PSS) with discovery of prediction rules underlying the prediction itself. He explored the use of C4.5 decision trees to extract relevant rules from PSS predictions modeled with two-stage support vector machines (TS-SVM). Dipankar Dasgupta delivered his keynote address on “Advances in Immunological Computation.” N.R. Pal’s talk on “Fuzzy Rule-Based Systems for Dimensionality Reduction” focused on the novelty of fuzzy rule-based systems used for dimensionality reduction through feature extraction preserving the “original structure” present in high-dimensional data. SEMCCO-2011 also included two invited talks and tutorial, which were free to all conference participants. The invited talks were delivered by Sumanth Yenduri, University of Southern Mississippi, and Amit Kumar, CEO and Chief Scientific Officer, Bio-Axis DNA Research Center, Hyderabad, on the topics “Wireless Sensor Networks—Sink Shift Algorithms to Maximize Efficiency” and “Eval-

VI

Preface

uating Mixed DNA Evidence with Forensic Bioinformatics,” respectively. The tutorial was delivered by Siba K. Udgata of the University of Hyderabad, India, on “Swarm Intelligence: An Optimization Tool for Various Engineering Applications.” The tutorial gave a brief overview of many swarm intelligence algorithms. The talk also covered an in-depth comparative study of these algorithms in different domains. In particular, emphasis was given to engineering applications like clustering in data mining, routing in networks, node placement in wireless sensor networks, finding the shortest path for packet forwarding, optimum resource allocation and planning, software failure prediction in software engineering, among many others. We take this opportunity to thank the authors of all submitted papers for their hard work, adherence to the deadlines and patience with the review process. The quality of a refereed volume depends mainly on the expertise and dedication of the reviewers. We are thankful to the reviewers for their timely effort and help rendered to make this conference successful. We are indebted to the Program Committee members who not only produced excellent reviews but also constantly encouraged us during the short time frames to make the conference of international repute. We would also like to thank our sponsors for providing all the support and financial assistance. First, we are indebted to ANITS Management and Administrations (The Secretary and Correspondent, the Principal and Directors and faculty colleagues and administrative personnel of the Departments of CSE, IT and MCA) for supporting our cause and encouraging us to organize the conference at ANITS, Vishakhapatnam. In particular, we would like to express our heart-felt thanks to Sri V. Thapovardhan, Secretary and Correspondent, ANITS, for providing us with the necessary financial support and infrastructural assistance to hold the conference. Our sincere thanks are due to V.S.R.K. Prasad, Principal, ANITS, for his continuous support. We thank Kalyanmoy Deb, IIT Kanpur, India, and Lakhmi Jain, Australia, for providing valuable guidelines and inspiration to overcome various difficulties in the process of organizing this conference as General Chairs. We extend our heart-felt thanks to Janusz Kacprzyk, Poland, for guiding us as the Honorary Chair of the conference. The financial assistance from ANITS and the others in meeting a major portion of the expenses is highly appreciated. We would also like to thank the participants of this conference, who have considered the conference above all hardships. Finally, we would like to thank all the volunteers whose tireless efforts in meeting the deadlines and arranging every detail ensured that the conference ran smoothly. December 2011

Bijaya Ketan Panigrahi Swagatam Das P.N. Suganthan Suresh Chandra Satapathy

Organization

Chief Patron Sri V.Thapovardhan

Secretary and Correspondent , ANITS

Patrons V.S.R.K. Prasad Govardhan Rao K.V.S.V.N. Raju

Principal, ANITS Director (Admin), ANITS Director (R & D), ANITS

Organizing Chairs S.C. Satapathy Ch. Suresh Ch. Sita Kameswari

HoD of CSE, ANITS HoD of IT, ANITS HoD of MCA, ANITS

Honorary Chair Janusz Kacprzyk

Poland

General Chairs Kalyanmoy Deb Lakhmi Jain

IIT Kanpur , India Australia

Program Chairs B.K. Panigrahi

Indian Institute of Technology (IIT), Delhi, India Swagatam Das Jadavpur University, Kolkata, India Suresh Chandra Satapathy ANITS, India

Steering Committee Chair P.N. Suganthan

Singapore

VIII

Organization

Publicity / Special Session Chair Sanjoy Das, USA Zhihua Cui, China Wei-Chiang Samuelson Hong, Taiwan

International Advisory Committee Almoataz Youssef Abdelaziz, Egypt Athanasios V. Vasilakos, Greece Boyang Qu, China Carlos A. Coello Coello, Mexico Chilukuri K. Mohan , USA Delin Luo, China Dipankar Dasgupta, USA Fatih M. Tasgetiren, Turkey Ferrante Neri, Finland G.K. Venayagamoorthy, USA Gerardo Beni, USA Hai Bin Duan, China Heitor Silv´erio Lopes, Brazil J.V.R. Murthy, India Jane J. Liang, China Janez Brest, Slovenia Jeng-Shyang Pan, Taiwan Juan Luis Fern´ andez Mart´ınez, USA K. Parsopoulos, Greece Kay Chen Tan, Singapore Leandro Dos Santos Coelho, Brazil Ling Wang, China Lingfeng Wang, China M.A. Abido, Saudi Arabia M.K. Tiwari, India Maurice Clerc, France Namrata Khemka, USA Oscar Castillo, Mexcico Pei-Chann Chang, Taiwan Peng Shi, UK P.V.G.D. Prasad Reddy, India Qingfu Zhang, UK Quanke Pan, China Rafael Stubs Parpinelli, Brazil Rammohan Mallipeddi, Singapore Roderich Gross, UK Ruhul Sarker, Australia

S. Baskar, India S.K. Udgata. India S.S. Dash, India S.S. Pattanaik, India S.G. Ponnambalam, Malaysia Saeid Nahavandi, Australia Saman Halgamuge, Australia Shizheng Zhao, Singapore X.Z. Gao, Finland Yew Soon Ong, Singapore Ying Tan, China Zong Wo Alex K. Qin, France Amit Konar, India Amit Kumar, India Anupam Shukla, India Ashish Anand, India Damodaram A., India D.K. Chaturvedi, India Dilip Pratihari, India Dipti Srinivasan, Singapore Frank Neumann, Australia G.S.N. Raju, India Hong Yan, Hong Kong Jeng-Shyang Pan, Taiwan John MacIntyre, UK Ke Tang, China M. Shashi, India Meng Joo Er., Singapore Meng-Hiot Lim, Singapore Oscar Castillo, Mexico P.K. Singh, India P.S. Avadhani, India Rafael Stubs Parpinelli, Brazil Richa Sing, India Robert Kozma, USA R. Selvarani, India Sachidananda Dehuri, India

Organization

Samuelson W. Hong, Taiwan Sumanth Yenduri, USA Suresh Sundaram, Singapore V. Kamakshi Prasad, India

V. Sree Hari Rao, India Yucheng Dong , China

Technical Review Board Clerc Maurice M. Willjuice Iruthayarajan Janez Brest Zhihua Cui Millie Pant Sidhartha Panda Ravipudi Rao Matthieu Weber Q.K. Pan Subramanian Baskar V. Ravikumar Pandi Krishnand K.R. Jie Wang V. Mukherjee S.P. Ghoshal Boyang Qu Tianshi Chen Roderich Gross Sanyou Zeng Ashish Ranjan Hota Yi Mei M. Rammohan Sambrata Dasg S. Miruna Joe Amali Kai Qin Bijan Mishra S. Dehury Shizheng Zhao Chilukuri Mohan Nurhadi Siswanto Aimin Zhou Nitin Anand Shrivastava Dipankar Maity Ales Zamuda Minlong Lin Ben Niu D.K. Chaturvedi Peter Koroˇsec

Mahmoud Abdallah Nidul Sinha Soumyadip Roy Anyong Qing Sanyou Zeng Siddharth pal Ke Tang Sheldon Hui Noha Hamza Kumar Gunjan Anna Kononova Noha Hamza Iztok Fister Fatih Tasgetiren Eman Samir Hasan Tianshi Chen Ferrante Neri Jie Wang Deepak Sharma Matthieu Weber Sayan Maity Abdelmonaem Fouad Abdallah Sheldon Hui Kenneth Price Nurhadi Siswanto S.N. Omkar Minlong Lin Shih-Hsin Chen Sasitharan Balasubramaniam Aniruddha Basak Shih-Hsin Chen Fatih Tasgetiren Soumyadip Roy S. Sivananaithapermal Borko Boskovic Pugalenthi Ganesan Ville Tirronen Jane Liang

IX

X

Organization

Ville Tirronen Bing Xue Andrea Caponio S. Sivananaithapermal Yi Mei Paramasivam Venkatesh Saber Elsayed

Saurav Ghosh Hamim Zafar Saber Elsayed Anyong Qing Arpan Mukhopadhyay Ye Xu

Organizing Committee P. Srinivasu B. Tirimula Rao M. James Stephen S. Ratan Kumar S. Jayaprada B. Ravi Kiran K. Neelima Santhoshi Ch. Demudu Naidu K.S. Deepthi Y.V. Srinivasa Murthy G. Jagadish G.V. Gayathri A. Kavitha A. Deepthi T. Kranthi S. Ranjan Mishra S.A. Bhavani K. Mrunalini S. Haleema M. Kranthi Kiran

K. Chandra Sekhar K. Sri Vaishnavi N. Sashi Prabha K. Santhi G. Gowri Pushpa K.S. Sailaja D. Devi Kalyani G. Santhoshi G.V.S. Lakshmi V. Srinivasa Raju Ch. Rajesh N. Sharada M. Nanili Tuveera Usha Chaitanya I. Sri Lalita Sarwani K. Yogeswara Rao T. Susan Salomi P. Lavanya Kumari K. Monni Sushma Deep S.V.S.S. Lakshmi

Table of Contents – Part I

Design of Two-Channel Quadrature Mirror Filter Banks Using Differential Evolution with Global and Local Neighborhoods . . . . . . . . . . . Pradipta Ghosh, Hamim Zafar, Joydeep Banerjee, and Swagatam Das

1

Differential Evolution with Modified Mutation Strategy for Solving Global Optimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pravesh Kumar, Millie Pant, and V.P. Singh

11

Self-adaptive Cluster-Based Differential Evolution with an External Archive for Dynamic Optimization Problems . . . . . . . . . . . . . . . . . . . . . . . . Udit Halder, Dipankar Maity, Preetam Dasgupta, and Swagatam Das

19

An Informative Differential Evolution with Self Adaptive Re-clustering Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dipankar Maity, Udit Halder, and Preetam Dasgupta

27

Differential Evolution for Optimizing the Hybrid Filter Combination in Image Edge Enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tirimula Rao Benala, Satchidananda Dehuri, G.S. Surya Vamsi Sirisetti, and Aditya Pagadala Scheduling Flexible Assembly Lines Using Differential Evolution . . . . . . . Lui Wen Han Vincent and S.G. Ponnambalam

35

43

A Differential Evolution Based Approach for Multilevel Image Segmentation Using Minimum Cross Entropy Thresholding . . . . . . . . . . . . Soham Sarkar, Gyana Ranjan Patra, and Swagatam Das

51

Tuning of Power System Stabilizer Employing Differential Evolution Optimization Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subhransu Sekhar Tripathi and Sidhartha Panda

59

Logistic Map Adaptive Differential Evolution for Optimal Capacitor Placement and Sizing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kamal K. Mandal, Bidishna Bhattacharya, Bhimsen Tudu, and Niladri Chakraborty Application of an Improved Generalized Differential Evolution Algorithm to Multi-objective Optimization Problems . . . . . . . . . . . . . . . . . Subramanian Ramesh, Subramanian Kannan, and Subramanian Baskar

68

77

XII

Table of Contents – Part I

Enhanced Discrete Differential Evolution to Determine Optimal Coordination of Directional Overcurrent Relays in a Power System . . . . . Joymala Moirangthem, Subranshu Sekhar Dash, K.R. Krishnanand, and Bijaya Ketan Panigrahi Dynamic Thinning of Antenna Array Using Differential Evolution Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ratul Majumdar, Aveek Kumar Das, and Swagatam Das A Quantized Invasive Weed Optimization Based Antenna Array Synthesis with Digital Phase Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ratul Majumdar, Ankur Ghosh, Souvik Raha, Koushik Laha, and Swagatam Das Optimal Power Flow for Indian 75 Bus System Using Differential Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aveek Kumar Das, Ratul Majumdar, Bijaya Ketan Panigrahi, and S. Surender Reddy

85

94

102

110

A Modified Differential Evolution Algorithm Applied to Challenging Benchmark Problems of Dynamic Optimization . . . . . . . . . . . . . . . . . . . . . . Ankush Mandal, Aveek Kumar Das, and Prithwijit Mukherjee

119

PSO Based Memetic Algorithm for Unimodal and Multimodal Function Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Swapna Devi, Devidas G. Jadhav, and Shyam S. Pattnaik

127

Comparison of PSO Tuned Feedback Linearisation Controller (FBLC) and PI Controller for UPFC to Enhance Transient Stability . . . . . . . . . . . M. Jagadeesh Kumar, Subranshu Sekhar Dash, M. Arun Bhaskar, C. Subramani, and S. Vivek

135

A Nelder-Mead PSO Based Approach to Optimal Capacitor Placement in Radial Distribution System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pradeep Kumar and Asheesh K. Singh

143

Comparative Performance Study of Genetic Algorithm and Particle Swarm Optimization Applied on Off-grid Renewable Hybrid Energy System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bhimsen Tudu, Sibsankar Majumder, Kamal K. Mandal, and Niladri Chakraborty

151

An Efficient Algorithm for Multi-focus Image Fusion Using PSO-ICA . . . Sanjay Agrawal, Rutuparna Panda, and Lingaraj Dora Economic Emission OPF Using Hybrid GA-Particle Swarm Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Preetha Roselyn, D. Devaraj, and Subranshu Sekhar Dash

159

167

Table of Contents – Part I

XIII

Application of Improved PSO Technique for Short Term Hydrothermal Generation Scheduling of Power System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Padmini, C. Christober Asir Rajan, and Pallavi Murthy

176

Multi-objective Workflow Grid Scheduling Based on Discrete Particle Swarm Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ritu Garg and Awadhesh Kumar Singh

183

Solution of Economic Load Dispatch Problem Using Lbest-Particle Swarm Optimization with Dynamically Varying Sub-swarms . . . . . . . . . . . Hamim Zafar, Arkabandhu Chowdhury, and Bijaya Ketan Panigrahi

191

Modified Local Neighborhood Based Niching Particle Swarm Optimization for Multimodal Function Optimization . . . . . . . . . . . . . . . . . Pradipta Ghosh, Hamim Zafar, and Ankush Mandal

199

Constrained Function Optimization Using PSO with Polynomial Mutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tapas Si, Nanda Dulal Jana, and Jaya Sil

209

Rank Based Hybrid Multimodal Fusion Using PSO . . . . . . . . . . . . . . . . . . . Amioy Kumar, Madasu Hanmandlu, Vaibhav Sharma, and H.M. Gupta

217

Grouping Genetic Algorithm for Data Clustering . . . . . . . . . . . . . . . . . . . . . Santhosh Peddi and Alok Singh

225

Genetic Algorithm for Optimizing Neural Network Based Software Cost Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tirimula Rao Benala, Satchidananda Dehuri, Suresh Chandra Satapathy, and Ch. Sudha Raghavi IAMGA: Intimate-Based Assortative Mating Genetic Algorithm . . . . . . . Fatemeh Ramezani and Shahriar Lotfi

233

240

SVR with Chaotic Genetic Algorithm in Taiwanese 3G Phone Demand Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Li-Yueh Chen, Wei-Chiang Hong, and Bijaya Ketan Panigrahi

248

Genetic Algorithm Assisted Enhancement in Pattern Recognition Efficiency of Radial Basis Neural Network . . . . . . . . . . . . . . . . . . . . . . . . . . . Prabha Verma and R.D.S. Yadava

257

An Approach Based on Grid-Value for Selection of Parents in Multi-objective Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rahila Patel, M.M. Raghuwanshi, and L.G. Malik

265

A Novel Non-dominated Sorting Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . Gaurav Verma, Arun Kumar, and Krishna K. Mishra

274

XIV

Table of Contents – Part I

Intelligent Genetic Algorithm for Generation Scheduling under Deregulated Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sundararajan Dhanalakshmi, Subramanian Kannan, Subramanian Baskar, and Krishnan Mahadevan

282

Impact of Double Operators on the Performance of a Genetic Algorithm for Solving the Traveling Salesman Problem . . . . . . . . . . . . . . . . . . . . . . . . . Goran Martinovic and Drazen Bajer

290

Parent to Mean-Centric Self-Adaptation in SBX Operator for Real-Parameter Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Himanshu Jain and Kalyanmoy Deb

299

Attribute Reduction in Decision-Theoretic Rough Set Models Using Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Srilatha Chebrolu and Sriram G. Sanjeevi

307

A Study of Decision Tree Induction for Data Stream Mining Using Boosting Genetic Programming Classifier . . . . . . . . . . . . . . . . . . . . . . . . . . . Dirisala J. Nagendra Kumar, J.V.R. Murthy, Suresh Chandra Satapathy, and S.V.V.S.R. Kumar Pullela Bi-criteria Optimization in Integrated Layout Design of Cellular Manufacturing Systems Using a Genetic Algorithm . . . . . . . . . . . . . . . . . . . I. Jerin Leno, S. Saravana Sankar, M. Victor Raj, and S.G. Ponnambalam

315

323

Reconfigurable Composition of Web Services Using Belief Revision through Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deivamani Mallayya and Baskaran Ramachandran

332

Neural Network Based Model for Fault Diagnosis of Pneumatic Valve with Dimensionality Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. Subbaraj and B. Kannapiran

341

A CAD System for Breast Cancer Diagnosis Using Modified Genetic Algorithm Optimized Artificial Neural Network . . . . . . . . . . . . . . . . . . . . . . J. Dheeba and S. Tamil Selvi

349

Application of ANN Based Pattern Recognition Technique for the Protection of 3-Phase Power Transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . Harish Balaga, D.N. Vishwakarma, and Amrita Sinha

358

Modified Radial Basis Function Network for Brain Tumor Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S.N. Deepa and B. Aruna Devi

366

Attribute Clustering and Dimensionality Reduction Based on In/Out Degree of Attributes in Dependency Graph . . . . . . . . . . . . . . . . . . . . . . . . . . Asit Kumar Das, Jaya Sil, and Santanu Phadikar

372

Table of Contents – Part I

MCDM Based Project Selection by F-AHP & VIKOR . . . . . . . . . . . . . . . . Tuli Bakshi, Arindam Sinharay, Bijan Sarkar, and Subir kumar Sanyal

XV

381

Nonlinear Time Series Modeling and Prediction Using Local Variable Weights RBF Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Garba Inoussa and Usman Babawuro

389

Detection of Disease Using Block-Based Unsupervised Natural Plant Leaf Color Image Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shitala Prasad, Piyush Kumar, and Anuj Jain

399

Measuring the Weight of Egg with Image Processing and ANFIS Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Payam Javadikia, Mohammad Hadi Dehrouyeh, Leila Naderloo, Hekmat Rabbani, and Ali Nejat Lorestani

407

Palmprint Authentication Using Pattern Classification Techniques . . . . . Amioy Kumar, Mayank Bhargava, Rohan Gupta, and Bijaya Ketan Panigrahi

417

A Supervised Approach for Gene Mention Detection . . . . . . . . . . . . . . . . . . Sriparna Saha, Asif Ekbal, and Sanchita Saha

425

Incorporating Fuzzy Trust in Collaborative Filtering Based Recommender Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vibhor Kant and Kamal K. Bharadwaj

433

A Function Based Fuzzy Controller for VSC-HVDC System to Enhance Transient Stability of AC/DC Power System . . . . . . . . . . . . . . . . . . . . . . . . Niranjan Nayak, Sangram Kesari Routray, and Pravat Kumar Rout

441

A Bayesian Network Riverine Model Study . . . . . . . . . . . . . . . . . . . . . . . . . . Steven Spansel, Louise Perkins, Sumanth Yenduri, and David Holt Application of General Type-2 Fuzzy Set in Emotion Recognition from Facial Expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anisha Halder, Rajshree Mandal, Aruna Chakraborty, Amit Konar, and Ramadoss Janarthanan

452

460

Design of a Control System for Hydraulic Cylinders of a Sluice Gate Using a Fuzzy Sliding Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wu-Yin Hui and Byung-Jae Choi

469

Rough Sets for Selection of Functionally Diverse Genes from Microarray Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sushmita Paul and Pradipta Maji

477

XVI

Table of Contents – Part I

Quality Evaluation Measures of Pixel - Level Image Fusion Using Fuzzy Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Srinivasa Rao Dammavalam, Seetha Maddala, and M.H.M. Krishna Prasad Load Frequency Control: A Polar Fuzzy Approach . . . . . . . . . . . . . . . . . . . Rahul Umrao, D.K. Chaturvedi, and O.P. Malik

485

494

An Efficient Algorithm to Computing Max-Min Post-inverse Fuzzy Relation for Abductive Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sumantra Chakraborty, Amit Konar, and Ramadoss Janarthanan

505

Fuzzy-Controlled Energy-Efficient Weight-Based Two Hop Clustering for Multicast Communication in Mobile Ad Hoc Networks . . . . . . . . . . . . . Anuradha Banerjee, Paramartha Dutta, and Subhankar Ghosh

520

Automatic Extractive Text Summarization Based on Fuzzy Logic: A Sentence Oriented Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Esther Hannah, T.V. Geetha, and Saswati Mukherjee

530

An Improved CART Decision Tree for Datasets with Irrelevant Feature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ali Mirza Mahmood, Mohammad Imran, Naganjaneyulu Satuluri, Mrithyumjaya Rao Kuppa, and Vemulakonda Rajesh Fuzzy Rough Set Approach Based Classifier . . . . . . . . . . . . . . . . . . . . . . . . . Alpna Singh, Aruna Tiwari, and Sujata Naegi Proposing a CNN Based Architecture of Mid-level Vision for Feeding the WHERE and WHAT Pathways in the Brain . . . . . . . . . . . . . . . . . . . . . Apurba Das, Anirban Roy, and Kuntal Ghosh Multithreaded Memetic Algorithm for VLSI Placement Problem . . . . . . . Subbaraj Potti and Sivakumar Pothiraj Bacterial Foraging Approach to Economic Load Dispatch Problem with Non Convex Cost Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Padmanabhan, R.S. Sivakumar, J. Jasper, and T. Aruldoss Albert Victoire Static/Dynamic Environmental Economic Dispatch Employing Chaotic Micro Bacterial Foraging Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nicole Pandit, Anshul Tripathi, Shashikala Tapaswi, and Manjaree Pandit Artificial Bee Colony Algorithm with Self Adaptive Colony Size . . . . . . . . Tarun Kumar Sharma, Millie Pant, and V.P. Singh

539

550

559

569

577

585

593

Table of Contents – Part I

Multi-Robot Box-Pushing Using Non-dominated Sorting Bee Colony Optimization Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pratyusha Rakshit, Arup Kumar Sadhu, Preetha Bhattacharjee, Amit Konar, and Ramadoss Janarthanan Emotion Recognition from the Lip-Contour of a Subject Using Artificial Bee Colony Optimization Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anisha Halder, Pratyusha Rakshit, Aruna Chakraborty, Amit Konar, and Ramadoss Janarthanan Software Coverage : A Testing Approach through Ant Colony Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bhuvnesh Sharma, Isha Girdhar, Monika Taneja, Pooja Basia, Sangeetha Vadla, and Praveen Ranjan Srivastava

XVII

601

610

618

Short Term Load Forecasting Using Fuzzy Inference and Ant Colony Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Amit Jain, Pramod Kumar Singh, and Kumar Anurag Singh

626

The Use of Strategies of Normalized Correlation in the Ant-Based Clustering Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Arkadiusz Lewicki, Krzysztof Pancerz, and Ryszard Tadeusiewicz

637

Ant Based Clustering of Time Series Discrete Data – A Rough Set Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Krzysztof Pancerz, Arkadiusz Lewicki, and Ryszard Tadeusiewicz

645

Sensor Deployment for Probabilistic Target k-Coverage Using Artificial Bee Colony Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Mini, Siba K. Udgata, and Samrat L. Sabat

654

Extended Trail Reinforcement Strategies for Ant Colony Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nikola Ivkovic, Mirko Malekovic, and Marin Golub

662

Fractional-Order PIλ Dμ Controller Design Using a Modified Artificial Bee Colony Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anguluri Rajasekhar, Vedurupaka Chaitanya, and Swagatam Das

670

Reconfiguration of Distribution Systems for Loss Reduction Using the Harmony Search Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.Y. Abdelaziz, Reham A. Osama, S.M. El-Khodary, and Bijaya Ketan Panigrahi An Improved Multi-objective Algorithm Based on Decomposition with Fuzzy Dominance for Deployment of Wireless Sensor Networks . . . . . . . . Soumyadip Sengupta, Md. Nasir, Arnab Kumar Mondal, and Swagatam Das

679

688

XVIII

Table of Contents – Part I

Application of Multi-Objective Teaching-Learning-Based Algorithm to an Economic Load Dispatch Problem with Incommensurable Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K.R. Krishnanand, Bijaya Ketan Panigrahi, P.K. Rout, and Ankita Mohapatra Application of NSGA – II to Power System Topology Based Multiple Contingency Scrutiny for Risk Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nalluri Madhusudana Rao, Diptendu Sinha Roy, and Dusmanta K. Mohanta Multi Resolution Genetic Programming Approach for Stream Flow Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rathinasamy Maheswaran and Rakesh Khosa

697

706

714

Reference Set Metrics for Multi-Objective Algorithms . . . . . . . . . . . . . . . . Chilukuri K. Mohan and Kishan G. Mehrotra

723

Groundwater Level Forecasting Using SVM-QPSO . . . . . . . . . . . . . . . . . . . Ch. Sudheer, Nitin Anand Shrivastava, Bijaya Ketan Panigrahi, and M Shashi Mathur

731

Genetic Algorithm Based Optimal Design of Hydraulic Structures with Uncertainty Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Raj Mohan Singh

742

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

751

Table of Contents – Part II

Register Allocation via Graph Coloring Using an Evolutionary Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sevin Shamizi and Shahriar Lotfi

1

A Survey on Swarm and Evolutionary Algorithms for Web Mining Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ashok Kumar Panda, S.N. Dehuri, M.R. Patra, and Anirban Mitra

9

Exploration Strategies for Learning in Multi-agent Foraging . . . . . . . . . . . Yogeswaran Mohan and S.G. Ponnambalam

17

Nurse Rostering Using Modified Harmony Search Algorithm . . . . . . . . . . . Mohammed A. Awadallah, Ahamad Tajudin Khader, Mohammed Azmi Al-Betar, and Asaju La’aro Bolaji

27

A Swarm Intelligence Based Algorithm for QoS Multicast Routing Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Manoj Kumar Patel, Manas Ranjan Kabat, and Chita Ranjan Tripathy Test Data Generation: A Hybrid Approach Using Cuckoo and Tabu Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Krish Perumal, Jagan Mohan Ungati, Gaurav Kumar, Nitish Jain, Raj Gaurav, and Praveen Ranjan Srivastava

38

46

Selection of GO-Based Semantic Similarity Measures through AMDE for Predicting Protein-Protein Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . Anirban Mukhopadhyay, Moumita De, and Ujjwal Maulik

55

Towards Cost-Effective Bio-inspired Optimization: A Prospective Study on the GPU Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Paula Prata, Paulo Fazendeiro, and Pedro Sequeira

63

Cricket Team Selection Using Evolutionary Multi-objective Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Faez Ahmed, Abhilash Jindal, and Kalyanmoy Deb

71

Data Clustering Using Harmony Search Algorithm . . . . . . . . . . . . . . . . . . . Osama Moh’d Alia, Mohammed Azmi Al-Betar, Rajeswari Mandava, and Ahamad Tajudin Khader

79

XX

Table of Contents – Part II

Application of Swarm Intelligence to a Two-Fold Optimization Scheme for Trajectory Planning of a Robot Arm . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tathagata Chakraborti, Abhronil Sengupta, Amit Konar, and Ramadoss Janarthanan

89

Two Hybrid Meta-heuristic Approaches for Minimum Dominating Set Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anupama Potluri and Alok Singh

97

Automatic Clustering Based on Invasive Weed Optimization Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aritra Chowdhury, Sandip Bose, and Swagatam Das

105

Classification of Anemia Using Data Mining Techniques . . . . . . . . . . . . . . . Shilpa A. Sanap, Meghana Nagori, and Vivek Kshirsagar

113

Taboo Evolutionary Programming Approach to Optimal Transfer from Earth to Mars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Mutyalarao, A. Sabarinath, and M. Xavier James Raj

122

Solving Redundancy Optimization Problem with a New Stochastic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chun-Xia Yang and Zhi-Hua Cui

132

Energy Efficient Cluster Formation in Wireless Sensor Networks Using Cuckoo Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Manian Dhivya, Murugesan Sundarambal, and J. Oswald Vincent

140

Data Clustering Based on Teaching-Learning-Based Optimization . . . . . . Suresh Chandra Satapathy and Anima Naik

148

Extracting Semantically Similar Frequent Patterns Using Ontologies . . . . S. Vasavi, S. Jayaprada, and V. Srinivasa Rao

157

Correlating Binding Site Residues of the Protein and Ligand Features to Its Functionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Ravindra Reddy, T. Sobha Rani, S. Durga Bhavani, Raju S. Bapi, and G. Narahari Sastry Non-linear Grayscale Image Enhancement Based on Firefly Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tahereh Hassanzadeh, Hakimeh Vojodi, and Fariborz Mahmoudi Synthesis and Design of Thinned Planar Concentric Circular Antenna Array - A Multi-objective Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sk. Minhazul Islam, Saurav Ghosh, Subhrajit Roy, Shizheng Zhao, Ponnuthurai Nagaratnam Suganthan, and Swagamtam Das

166

174

182

Table of Contents – Part II

Soft Computing Based Optimum Parameter Design of PID Controller in Rotor Speed Control of Wind Turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . R. Manikandan and Nilanjan Saha Curve Fitting Using Coevolutionary Genetic Algorithms . . . . . . . . . . . . . . Nejat A. Afshar, Mohsen Soryani, and Adel T. Rahmani A Parallel Hybridization of Clonal Selection with Shuffled Frog Leaping Algorithm for Solving Global Optimization Problems (P-AISFLA) . . . . . Suresh Chittineni, A.N.S. Pradeep, G. Dinesh, Suresh Chandra Satapathy, and P.V.G.D. Prasad Reddy Non-uniform Circular-Shaped Antenna Array Design and Synthesis - A Multi-Objective Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saurav Ghosh, Subhrajit Roy, Sk. Minhazul Islam, Shizheng Zhao, Ponnuthurai Nagaratnam Suganthan, and Swagatam Das Supervised Machine Learning Approach for Bio-molecular Event Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Asif Ekbal, Amit Majumder, Mohammad Hasanuzzaman, and Sriparna Saha Design of Two Channel Quadrature Mirror Filter Bank: A Multi-Objective Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subhrajit Roy, Sk. Minhazul Islam, Saurav Ghosh, Shizheng Zhao, Ponnuthurai Nagaratnam Suganthan, and Swagatam Das Soft Computing Approach for Location Management Problem in Wireless Mobile Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Moumita Patra and Siba K. Udgata Distribution Systems Reconfiguration Using the Hyper-Cube Ant Colony Optimization Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.Y. Abdelaziz, Reham A. Osama, S.M. El-Khodary, and Bijaya Ketan Panigrahi

XXI

191

201

211

223

231

239

248

257

Bacterial Foraging Optimization Algorithm Trained ANN Based Differential Protection Scheme for Power Transformers . . . . . . . . . . . . . . . . M. Geethanjali, V. Kannan, and A.V.R. Anjana

267

Reduced Order Modeling of Linear MIMO Systems Using Soft Computing Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Umme Salma and K. Vaisakh

278

Statistical and Fusion Based Hybrid Approach for Fault Signal Classification in Electromechanical System . . . . . . . . . . . . . . . . . . . . . . . . . . Tribeni Prasad Banerjee and Swagatam Das

287

XXII

Table of Contents – Part II

Steganalysis for Calibrated and Lower Embedded Uncalibrated Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deepa D. Shankar, T. Gireeshkumar, and Hiran V. Nath

294

An Efficient Feature Extraction Method for Handwritten Character Recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Manju Rani and Yogesh Kumar Meena

302

Optimized Neuro PI Based Speed Control of Sensorless Induction Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. Arulmozhiyal, C. Deepa, and Kaliyaperumal Baskaran

310

Wavelet Based Fuzzy Inference System for Simultaneous Identification and Quantitation of Volatile Organic Compounds Using SAW Sensor Transients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Prashant Singh and R.D.S. Yadava Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

319

329

Design of Two-Channel Quadrature Mirror Filter Banks Using Differential Evolution with Global and Local Neighborhoods Pradipta Ghosh, Hamim Zafar, Joydeep Banerjee, and Swagatam Das Electronics & Tele-Comm. Engineering Jadavpur University Kolkata, India {iampradiptaghosh,hmm.zafar,iamjoydeepbanerjee}@gmail.com, [email protected]

Abstract. This paper introduces a novel method named DEGL (Differential Evolution with global and local neighborhoods) regarding the design of two channel quadrature mirror filter with linear phase characteristics. To match the ideal system response characteristics, this improved variant of Differential Evolution technique is employed to optimize the values of the filter bank coefficients. The filter response is optimized in both pass band and stop band. The overall filter bank response consists of objective functions termed as reconstruction error, mean square error in pass band and mean square error in stop band. Effective designing can be performed if the objective function is properly minimized. The proposed algorithm can perform much better than the other existing design methods. Three different design examples are presented here for the illustrations of the benefits provided by the proposed algorithm. Keywords: Filter banks, Quadrature Mirror Filter, Sub-band coding, perfect reconstruction, DEGL.

1 Introduction Efficient design of filter banks has become a promising area of research work. An improved design of filter can have significant effect on different aspects of signal processing and many fields such as speech coding, scrambling, image processing, and transmission of several signals through same channel [1]. Among various filter banks, the two channel QMF bank was first used in Sub-band coding, in which the signal is divided into several frequency bands and digital encoders of each band of signal can be used for analysis. QMF also finds application in representation of signals in digital form for transmission and storage purpose in speech processing, image processing and its compression, communication systems, power system networks, antenna systems [2], analog to digital (A/D) converter [3], and design of wavelet base [4]. Various optimizations based or non optimization based design techniques for QMF have been found in literature. Recently several efforts have been made for designing the optimized QMF banks based on linear and non-linear phase objective function using various evolutionary algorithms. Various methods such as least square technique [5-7], B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 1–10, 2011. © Springer-Verlag Berlin Heidelberg 2011

2

P. Ghosh et al.

weighted least square (WLS) technique [8-9] have been applied to solve the problem. But due to high degree of nonlinearity and complex optimization technique, these methods were not suitable for the filter with larger taps. A method based on eigenvector computation in each iteration is proposed to obtain the optimum quantized filter weights [10]. An optimization method based on Genetic algorithm and signed-power-of-two [11] is successfully applied in designing the lattice QMF. In frequency domain methods reconstruction error is not equiripple [7-9]. Chen and Lee have proposed an iterative technique [8] that results in equiripple reconstruction error, and the generalization of this method was carried out in [9] to obtain equiripple behaviors in stop band. Unfortunately, these techniques are complicated, and are only applicable to the two-band QMF banks that have low orders. To solve the previous problems, a two-step approach for the design of two-channel filter banks [12,13] was developed. But this approach results in nonlinear phase, and is not suitable for the wideband audio signal. A more robust and powerful tool PSO has also been applied for the design of the optimum QMF bank with reduced aliasing distortion, phase aliasing and reconstruction errors [14,15]. The problem with PSO is the premature convergence due to the presence of local optima. Amongst all Evolutionary Algorithms (EAs) described in various articles, Differential Evolution (DE) has emerged as one of the most powerful tools for solving the real world optimization problems. It has not been applied on the design of the perfectly reconstructed QMF banks till now. In this context we present here a new powerful variant of DE called DEGL [20] for the efficient design of two channels QMF bank. Later in this paper, we will discuss the effectiveness of this algorithm and compare this with other existing method towards designing of Digital QMF filter. For proving our point we have presented three different design problems. The rest of the paper is arranged in the following way: Section 2 contains the Design Problem, Section 3 gives a brief overview of classical DE, Section 4 introduces a new improved variant of DE termed as DEGL, Section 5 deals with the design results and comparison of these results with other algorithms and Section 6 concludes this paper.

2 Formulation of Design Problems For a typical two-channel QMF bank as shown in Fig. 1 the reconstructed output signal is defined as Y ( z) = 1/ 2[ H 0 ( z)G0 ( z) + H1 ( z)G1 ( z)]X ( z) + 1/ 2[ H 0 (− z)G0 ( z) + H1 (− z)G1 ( z)]X (− z)

= T ( z ) X ( z ) + A( z ) X ( − z )

where Y(z) is the reconstructed signal and X(z) is the original signal.

Fig. 1. Two channel QMF BANK

(1)

Design of Two-Channel Quadrature Mirror Filter Banks

3

In eqn.1 the first term T(z) is the desired translation from input to the output, called distortion transfer function, while second term, A(z) is aliasing distortion because of change in sampling rate. By setting

H 1 ( z ) = H 0 (− z ), G0 ( z ) = H 1 (− z ) and G1 ( z ) = − H 0 ( − z )

(2)

the aliasing error in (1) can be completely eliminated. Then eqn. (1) becomes

Y ( z) = 1/ 2[ H 0 ( z) H 0 ( z) + H 0 (− z) H 0 (− z)] X ( z )

(3)

It implies that the overall design problem of filter bank reduces to determination of the filter taps coefficients of a low pass filter

H 0 (z) , called a prototype filter. Let H 0 (z) be a linear phase finite impulse response (FIR) filter with even length ( N ): H 0 (e jω ) = H 0 (e jω ) e − jω ( N −1) / 2

(4)

If all above mentioned conditions are put together, the overall transfer function of − jω ( N −1) / 2 2 2 QMF bank is reduced to Eqn.(5) T (e jω ) = e (5) H (e jω ) + H ( e j ( ω − π ) 2

T ( e jω ) =

{

0

0

}

1 − jω ( N −1) / 2 e T ′(e jω ) 2

(6)

Where, T ′(e jω ) = [ H 0 (ω ) 2 − (−1)( N −1) H 0 (π − ω ) 2 ] and N is the no of filters. If

2

2

H 0 (e jω ) + H 0 (e j (ω −π ) = 1 it results in perfect reconstruction; the output signal

is exact replica of the input signal. If it is evaluated at frequency ( ω = 0.5π ), the perfect reconstruction condition reduces to Eqn. (7). H 0 (e j 0.5π ) = 0.707

(7) − jω ( N −1) / 2

It shows that QMF bank has a linear phase delay due to the term e . Then the condition for the perfect reconstruction is to minimize the weighted sum of four terms as shown below:

K = α1 E p + α 2 E s + α 3 Et + α 4 .mor + α 5 E h

(8)

where α 1 − α 5 are the relative weights and E p , E s , E t , mor (measure of ripple) ,

Eh are defined as follows: E p is the mean square error in pass band (MSEP) which describes the energy of reconstruction error between 0 and Ep =

1

π

ω

H

(0) − H 0 (ω ) dω 2

0

ωp , (9)

0

E s is the mean square error in stop band (MSES) which denotes the stop band energy related to LPF between

ω s to π

4

P. Ghosh et al.

Es =

1

π

π

H

(10)

(ω ) dω 2

0

ωs

E t is the square error of overall transfer function at quadrature frequency π / 2 π 1 Et = [ H 0 ( ) − H 0 (0)]2 2 2

(11)

mor = max 10 log 10 T ′(ω ) − min 10 log 10 T ′(ω )

(12)

Measure of ripple ( mor ) ω

ω

Eh is the deviation of T ′(e jω ) from unity at ω = π / 2 π

(13)

E h = T ′( ) − 1 2

The above fitness is considered because we know that condition for perfect reconstruction filter is 2

2

H 0 (e jω ) + H 0 (e j (ω −π ) = 1

(14)

3 DE with Global and Local Neighborhoods (DEGL) DE is a simple real-coded evolutionary algorithm [16]. It works through a simple cycle of stages, which are detailed in [16]. In this section we describe the variant of DE termed as DEGL. Suppose we have a DE population

    PG = [ X 1,G , X 2,G , X 3,G ,....., X NP ,G ] 

Where each X i ,G

(15)

(i = 1,2,3,......, NP ) a D dimensional parameter vector. Now, for

 every vector X i ,G we define a neighborhood of radius k (where k is a nonzero integer

from 0 to (NP-1)/2, as the neighborhood size must be smaller than the population size,    i.e. 2k + 1 ≤ NP), consisting of vectors X i − k ,G ,. . , X i ,G ,…, X i + k ,G . We assume the vectors to be organized on a ring topology with respect to their indices, such that vec   tors X NP ,G and X 2 ,G are the two immediate neighbors of vector X 1,G .For each member of the population, a local donor vector is created by employing the best (fittest) vector in the neighborhood of that member and any two other vectors chosen from the same neighborhood. The model may be expressed as

      Li ,G = X i ,G + α .( X n _ besti ,G − X i ,G ) + β .( X p ,G − X q ,G ) (16)

Design of Two-Channel Quadrature Mirror Filter Banks

5

where the subscript n_besti indicates the best vector in the neighborhood of X i ,G and p, q ∈[i - k, i + k] with p ≠ q ≠ i . Similarly, the global donor vector is created as

      g i ,G = X i ,G + α .( X n _ gbest ,G − X i ,G ) + β .( X r1,G − X r 2,G )

(17)

where the subscript g_best indicates the best vector in the entire population at generation G and r 1 , r 2 ∈ [ 1 , NP ] with r1 ≠ r 2 ≠ i . α and β are the scaling factors. Note that in (16) and (17), the first perturbation term on the right-hand side (the one multiplied by α ) is an arithmetical recombination operation, while the second term (the one multiplied by β ) is the differential mutation. Thus in both the global and local mutation models, we basically generate mutated recombinants, not pure mutants. Now we combine the local and global donor vectors using a scalar weight ω ∈ ( 0 ,1 ) to form the actual donor vector of the proposed algorithm

   Vi ,G = ω . g i ,G + (1 − ω ).. Li ,G

(18)

Clearly, if ω = 1 and in addition α = β = F , the donor vector generation scheme in (18) reduces to that of DE/target to-best/1. Hence the latter may be considered as a special case of this more general strategy involving both global and local neighborhood of each vector synergistically. From now on, we shall refer to this version as DEGL (DE with global and local neighborhoods). The rest of the algorithm is exactly similar to DE/rand/1/bin. DEGL uses a binomial crossover scheme. 3.1 Control Parameters in DEGL DEGL introduces four new parameters. They are: α , β , ω and the neighborhood radius k. In order to lessen the number of parameters further, we take α = β = F . The most important parameter in DEGL is perhaps the weight factor ω , which controls the balance between the exploration and exploitation capabilities. Small values of ω (close to 0) in (11) favor the local neighborhood component, thereby resulting in better exploration. There are three different schemes for the selection and adaptation of ω to gain intuition regarding DEGL performance. They are Increasing weight Factor, Random Weight Factor, Self-Adaptive Weight Factor respectively. But we have used only Random Weight Factor for this design problem. So we will describe only the incorporated method in the following paragraphs. 3.2 Random Weight Factor In this scheme the weight factor of each vector is made to vary as a uniformly distributed random number in (0, 1) i.e. ω i ,G ≈ rand (0,1) . Such a choice may decrease the convergence speed (by introducing more diversity). But the minimum value is 0.15.

6

P. Ghosh et al.

3.3 Advantage of Random Weight Factor This scheme had empirically proved to be the best scheme among all three schemes defined in original DEGL article for this kind of design problem. The most important advantage in this scheme lies on the process of crossover. Due to varying weight factor the no of possible different vector increases. So the searching is much wider than using other two schemes.

4 Design Problems 4.1 Parameter Initializations For the design purpose we set the searching upper bound = 0.5 and searching lower bound = -0.5; Function bound Constraint for DEGL is set to be 0. The initial population size is 100. The no of generations for DEGL is set equal to 500. Next we had to set the values of the constant terms i.e. α1 − α 5 in Eqn. 8. For all the examples, the relative weights of fitness function are determined based on trial and error method using concepts of QMF filter. The values of the constants are as follows. α1 = .95, α 2 = .07, α 3 = .07, α 4 = 10 −4 , α 5 = 10 −1. 4.2 Problem Examples 4.2.1 Two-Channel QMF Bank for N = 22, ω p = 0.4π , ω s = 0.6π , with 11 Filter Coefficient, h0 → h10 . For filter length N = 22,

ω p = 0.4π

, edge frequency of stop-band

ω s = 0.6π . The

normalized amplitude response for H0, H1 filters of analysis bank and amplitude of distortion function T ′(ω ) are plotted in Figs. 2(a) and 2(b), respectively. From Fig. 2(c), this represents attenuation characteristic of low-pass filter H0. Fig. 2(d) represents the reconstruction error of QMF bank. Table 1 provides the filter characteristics. h0 = -0.00161 h1 = -0.00475 h2 = 0.01330 h3 = 0.00104 h4 = -0.02797

h5 = 0.00940 h6 = 0.05150 h10 = 0.46861. 4.2.2

h7 = -0.03441 h8 = -0.10013 h9 = 0.12490

Two-Channel QMF Bank for N = 32, ω p = 0.4π , ω s = 0.6π , with 16 Filter Coefficient, h0 → h16 .

For filter length N = 32, ω p = 0.4π , edge frequency of stop-band

ω s = 0.6π . After

setting the initial values and parameters, the DEGL algorithm is run to obtain the optimal filter coefficients. The normalized amplitude response for H0, H1 filters of analysis bank and amplitude of distortion function T ′(ω ) are plotted in Figs. 4(a)

Design of Two-Channel Quadrature Mirror Filter Banks

(a)

(b)

(c)

(d)

7

Fig. 2. The frequency response of Example 2. (a) Normalized amplitude response of analysis bank. (b) Amplitude of distortion function. (c) Low-pass filter attenuation characteristics in dB. (d) Reconstruction error in dB, for N =22.

(a)

(b)

(c)

(d)

Fig. 3. The frequency response of Example 3. (a) Normalized amplitude response of analysis bank. (b) Amplitude of distortion function. (c) Low-pass filter attenuation characteristics in dB. (d) Reconstruction error in dB, for N =32.

8

P. Ghosh et al.

and 4(b), respectively. From Fig. 4(c), this represents attenuation characteristic of low-pass filter H0. Fig. 4(d) represents the reconstruction error of QMF bank. Table 2 presents a comparison of proposed DEGL method with design based on CLPSO and DE algorithms. Table 1 provides the filter characteristics. The optimized 16-filter coefficients for the analysis bank low-pass filter are given below. h0 = 0.0012; h1 = -0.0024; h2 = -0.0016; h3 = 0.0057; h4 = 0.0013; h5 = -0.0115

h6 = 0.0008; h7 = 0.0201; h8 = -0.0058; h9 = -0.0330; h10 = 0.0167; h11 = 0.0539 h12 = -0.0417; h13 = -0.0997; h14 = 0.1306 ; h15 = 0.4651 Tab1e 1. Filter’s performance measuring quantities

SBEA (stop-band edge attenuation ) SBLFA ( stop-band first lobe attenuation ) MSEP( mean square error in pass band ) MSES ( mean square error in stop band ) mor (measure of ripple)

N=22 20.8669 dB 31.1792 dB 8.95 × 10 = 07

N=32 34.7309 dB 44.6815 dB 1.58 × 10 = 07

1.23 × 10= 04 0.0186

3.15 × 10= 06 0.0083

Table 2. Performance comparison of proposed DEGL method with other algorithms Name of the algorithm CL- PSO[19] DE[16] DEGL

mor 0.0256 0.0123 0.0083

Filter constants and parameters MSEP MSES SBEA 23.6382 1.12 ×10 = 03 1.84 ×10 = 04 9.72 × 10 = 06 6.96 × 10= 04 29.6892 = 07 1.58 × 10 3.15 × 10= 06 34.7309

SBLFA 31.9996 39.8694 44.6815

5 Discussion of Results We have described here two design problems with no of filter coefficients equal to 11 and 16. One comparison table is also given for the 2nd design problem. In Table 2 the results for N=32 are compared with the results of same design problem using CL-PSO [19] and DE [16] algorithms. The results clearly show that this method leads to filter banks with improved performance in terms of peak reconstruction error, mean square error in stop band, pass band. The measure of ripple, which is an important parameter in signal processing, is also considerably lower than DE and CLPSO based methods for all the problems. Almost all the problems satisfy Eqn. 13 which is the condition for perfect reconstruction filter which is one of the achievements of this algorithm. The corresponding values of MSEP and MSES are also much lower in this method than the other two methods as shown in table 4. Also DEGL based method is much better in terms SBEA and SBLFA.

Design of Two-Channel Quadrature Mirror Filter Banks

9

6 Conclusions In this paper, a DEGL Algorithm based technique is used for the design of QMF bank Simulation The result shows that design of filter using DEGL is very effective and efficient for QMF filter design for any number of filter coefficients. We could also use other improved DE algorithms and many other evolutionary algorithms for design purpose. We can also use multi objective algorithms and multimodal optimization algorithms [21]. Our further works will be focused on the improvement of the result obtained using some new design scheme and further optimization techniques.

References 1. Vaidyanathan, P.P.: Multirate Systems and Filter Banks. Prentice Hall Inc. (1993) 2. Chandran, S., Ibrahim, M.K.: Adaptive antenna techniques using QMF bank. In: IEEE International Conference on Antennas and Propagation, pp. 257–260 (April 1995) 3. Petraglia, A., Mitra, S.K.: High speed A /D conversion incorporating a QMF. IEEE Trans. on Instrumentation and Measurement 41(3), 427–431 (1992) 4. Chan, S.C., Pun, C.K.S., Ho, K.L.: New design and realization techniques for a class of perfect reconstruction two-channel FIR filter banks and wavelet bases. IEEE Transactions Signal Processing 52(7), 2135–2141 (2004) 5. Johnston, J.D.: A filter family designed for use in quadrature mirror filter banks. In: Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing, pp. 291–294 (April 1980) 6. Pirani, G., Zingarelli, V.: An analytical formula for the design of quadrature mirror filters. IEEE Transactions Acoustic, Speech, Signal Processing ASSP-32, 645–648 (1984) 7. Jain, V.K., Crochiere, R.E.: Quadrature mirror filter design in time domain. IEEE Transactions on Acoustics Speech and Signal Processing ASSP-32, 353–361 (1984) 8. Chen, C.K., Leem, J.H.: Design of quadrature mirror filters with linear phase in the frequency domain. IEEE Transactions on Circuits and Systems II 39(9), 593–605 (1992) 9. Lim, Y.C., Yang, R.H., Koh, S.N.: The design of weighted minimax quadrature mirror filters. IEEE Transactions Signal Processing 41(5), 1780–1789 (1993) 10. Andrew, L., Franques, V.T., Jain, V.K.: Eigen design of quadrature mirror fil-ters. IEEE Trans. Circuits Syst. II Analog Digit.Signal Process. 44(9), 754–757 (1997) 11. Yu, Y.J., Lim, Y.C.: New natural selection process and chromosome encoding for the design of multiplier less lattice QMF using genetic algorithm. In: 8th IEEE International Conf. Electronics, Circuits and Systems, vol. 3, pp. 1273–1276 (2001) 12. Bregovic, R., Saramaki, T.: Two-channel FIR filter banks - A tutorial review and new results. In: Proceeding Second International Workshop on Transforms and Filter Banks, Brandenburg, Germany (March 1999) 13. Bregovic, R., Saramaki, T.: A general-purpose optimization approach for design-ing twochannel FIR filter banks. IEEE Transactions on Signal Processing 51(7) (July 2003) 14. Kumar, A., Singh, G.K., Anand, R.S.: Design of Quadrature Mirror Filter Bank us-ing Particle Swarm Optimization (PSO). International Journal of Recent Trends in Engineering 1(3) (May 2009) 15. Upendar, J., Gupta, C.P., Singh, G.K.: Design of two-channel quadrature mirror filter bank using particle swarm optimization. Elsevier Science Direct Digital Signal Processing 20, 304–313 (2010) 16. Storn, R., Price, K.: Differential evolution a simple and efficient heuristic for global optimization over continuous spaces. J. Global Optimization 11(4), 341–359 (1997)

10

P. Ghosh et al.

17. Mendes, R., Kennedy, J.: The fully informed particle swarm: Simpler may be better. IEEE Trans. Evol. Comput. 8(3), 204–210 (2004) 18. Storn, R., Price, K.V.: Differential Evolution - a simple and efficient adaptive scheme for global optimization over continuous spaces, Technical Report TR-95-012, ICSI (1995), http://http.icsi.berkeley.edu/~storn/litera.html 19. Liang, J.J., Qin, A.K., Suganthan, P.N., Baskar, S.: Comprehensive learning particle swarm optimizer for global optimization of multimodal functions. IEEE Trans. Evol. Comput. 10(3), 281–295 (2006) 20. Das, S., Abraham, A., Chakraborty, U.K., Konar, A.: Differential Evolution Using a Neighborhood-Based Mutation Operator. IEEE Transactions on Evolutionary Computation 13(3), 526–553 (2009) 21. Qu, B.Y., Suganthan, P.N., Liang, J.J.: Differential Evolution with Neighborhood Mutation for Multimodal Optimization. IEEE Trans. on Evolutionary Computation, doi:10.1109/TEVC.2011.2161873

Differential Evolution with Modified Mutation Strategy for Solving Global Optimization Problems Pravesh Kumar1, Millie Pant1, and V.P. Singh2 1

2

Indian Insitute of Technology, Roorkee, India Millenium Insitute of Engineering and Technolgy, India [email protected], [email protected]

Abstract. In the present work we propose a modified variant of Differential Evolution (DE) algorithm named MDE. MDE differs from the basic DE in the manner in which the base vector is generated. While in simple/basic DE, base vector is usually randomly selected from the population of individuals, in MDE base vector is generated as convex linear combination (clc) of three randomly selected vectors out of which one is the one having best fitness value. This mutation scheme is used stochastically with mutation scheme in which the base generated using a clc of three randomly generated vectors. MDE is validated on a set of benchmark problems and is compared with basic DE and other DE variants. Numerical and statistical analysis shows the competence of proposed MDE. Keywords: differential evolution, mutation strategy, global optimization.

1

Introduction

Differential Evolution (DE) algorithm is an evolutionary algorithm, which was proposed by Storn and Price in 1995 [1]. Differential evolution (DE) [1], [2], is a simple and efficient, stochastic, population set based methods for global optimization over continuous space. DE is capable of handling non-differentiable, nonlinear and multimodal objective functions and have been applied successfully to a wide range of problem such as real-world applications, such as data mining [3], [4], pattern recognition, digital filter design, neural network training, etc. [5], Compared with Particle Swarm Optimization (PSO) and Genetic Algorithm (GA), DE algorithm has many advantages, such as faster convergence speed, stronger stability, easy to realize [6] and so on, so it is noticed by many researchers. In order to improve the performance of DE, its several variants have been proposed. There are various mutation strategies available in the literature [7]-[11]. In this paper we have taken the basic DE strategy DE/rand/1/bin [8], [9]. We shall refer to it as basic/ simple DE (SDE). In the present study we have proposed a new variant of DE, named MDE using a modified mutation strategy, based on [10]. The rest of paper is organized as follows: Section 2 provides a compact overview of basic DE. Section 3 presents the proposed MDE algorithm. Numerical Simulation and Comparisons are reported in Section 4, and finally the conclusions derived from the present study are drawn in Section 5. B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 11–18, 2011. © Springer-Verlag Berlin Heidelberg 2011

12

2

P. Kumar, M. Pant, and V.P. Singh

Basic DE

Basic DE algorithm is a kind of evolutionary algorithm, used for function optimization [1]. The structure of DE is similar to the GA and both use the operators’ selection, mutation and crossover to guide the search process. The main difference between standard GA and DE is mutation operation. Mutation is a main operation of DE, and it revises each individual’s value according to the difference vectors of the population. The algorithm uses mutation operation as a search mechanism; crossover operation to induce diversity and selection operation to direct the search toward the potential regions in the search space. The working of DE is as follows: First, all individuals are initialized with uniformly distributed random numbers and evaluated using the fitness function provided. Then the following are executed until maximum number of generation has been reached or an optimum solution is found. Mutation: For a D-dimensional search space, for each target vector X i,G at the generation G, its associated mutant vector is generated via certain mutation strategy. The most often used mutation strategy implemented in the DE is given by equation-1. DE/rand/1/bin: Vi ,G+1 = X r1 ,G + F * ( X r2 ,G − X r3 ,G ) where

(1)

r1 , r2 , r3 ∈ {1,2,...., NP} are randomly chosen integers, different from each

other and also different from the running index i. F (>0) is a scaling factor which controls the amplification of the difference vectors. Crossover: Once the mutation phase is over, crossover is performed between the target vector and the mutated vector to generate a trial point for the next generation. The mutated individual, Vi,G+1 = (v1,i,G+1, . . . , vD,i,G+1), and the current population member ( target vector), Xi,G = (x1,i,G, . . . , xD,i,G), are then subject to the crossover operation, that finally generates the population of candidates, or “trial” vectors,Ui,G+1 = (u1,i,G+1, . . . , uD,i,G+1), as follows:

v j ,i ,G +1 u j ,i ,G +1 =   x j ,i ,G

if

rand j ≤ Cr ∨ j = k otherwise

(2)

where j, k ∈ {1,…, D} k is a random parameter index, chosen once for each i, Cr is the crossover probability parameter whose value is generally taken as Cr ∈ [0, 1] . Selection: The final step in the DE algorithm is the selection process. Each individual of the temporary (trial) population is compared with the corresponding target vector in the current population. The one with the lower objective function value survives the tournament selection and goes to the next generation. As a result, all the individuals of the next generation are as good as or better than their counterparts in the current generation.

Differential Evolution with Modified Mutation Strategy

13

f (U i ,G +1 ) ≤ f ( X i ,G ) otherwise

(3)

U i ,G +1 X i ,G +1 =   X i ,G

3

if

Proposed Modified DE (MDE)

In this section, we describe the proposed MDE. Here, we used two new mutation strategies: first strategy (say S1) is selected from [10] and the second strategy (say S2) is novel strategy. Both S1 and S2, differ from the basic DE mutation strategy in the selection of base vector. While in basic DE, a single vector is selected as a base vector in S1 and S2, the base vector is a convex linear combination of three randomly selected vectors of the population. In S1, the three vectors are randomly selected while in S2, only two vectors are randomly selected and the third vector is the one having best fitness value X best , G The strategies, S1 and S2, are defined as

S1 : Vi , G +1 = ( μ1 X r1, G + μ 2 X r 2, G + μ 3 X r 3, G ) + F ( X r 2, G − X r 3, G ) S 2 : Vi , G +1 = ( μ1 X best , G + μ 2 X r 2, G + μ 3 X r 3, G ) + F ( X r 2, G − X r 3, G )

(4)

Here μi i=1, 2 are uniform random number between 0 and 1 and μ3=1-( μ1 +μ2), (satisfies the condition

3

μ i =1

i

= 1 ).

These strategies are stochastically applied using a fixed probability (Pr). A random number (R) is generated uniformly between 0 & 1. If the value of R is less than Pr then S1 is selected otherwise is S2 selected. 3.1

Pseudo Code for MDE

Begin Create uniformly random population Xi,G, where i=1,2,..,NP while(termination criteria is met) for i=1:NP { Select three random parents Xr1,G, Xr2,G , Xr3,G and Xbest,G from the current population where i≠r1≠r2≠r3 { if(Pr f ( X i ,t )

for minimization problem. Thus after every generation we find either a new solution which has better fitness (here for minimization problem) or the previous vector is kept. So after each generation the population gets better or remains unchanged but never deteriorates.

3 Proposed Algorithm: SACDEEA In this paper we propose a strategy based, self-adaptive, clustered DE with an External Archive to solve the DOPs. Here, in this algorithm we divide the whole population in several clusters based on their spatial positions. Before describing the algorithm in details let us define some terms regarding the clusters.

    x1 , x2 ….., xm then the center ( C ) is determined as:

• If the particles of a cluster are

 xi

m



 C =

i =1

(3)

m

• The radius (R) of a cluster is defined as the mean distance (Euclidean) of the particles from the center of the cluster. So, we can write m

n

i =1

j =1

  (x R

=

i, j

m where n is the dimension of the problem.

− C j )2 ;

(4)

• For any cluster, we define a fixed value, named convergence radius ( Rconv ), which is calculated as

22

U. Halder et al.

  R conv = dist ( X upper , X lower ) ⋅ 10 − 3

,

(5)

  where Xupper and Xlower are the upper bound and lower bound vectors in the search    region, and d is t ( a , b ) denotes the Euclidean distance between the vectors a and b . • Every cluster can contain up to a maximum number (pop_max) of individuals. It has been seen that DE performs better than other EAs in many optimization problems. So, we used DE as our main improvement algorithm with some enhancements to handle dynamic problems. The modifications we have made are described below. 3.1 Initialization

A certain number of individuals (NP) are spread randomly in the search region. Then they are clustered into k clusters using the K-means clustering algorithm. 3.2 Cluster Improvements

After clustering is done, evolutionary strategy is used to improve the members of the cluster. For this purpose, “DE/best/1” scheme (2) is used separately in each cluster, i.e. there is no information sharing among the clusters in a specific generation. This might prevent the algorithm to converge all the particles to a single local optimum or in other words, premature convergence is discouraged. The clustering technique is used to explore the search region more and search for better and promising areas simultaneously instead of optimizing a single population and therefore guiding all the particles to converge to one optimum, which may be a false one. By this way, the diversity of the population can be maintained. 3.3 Performance Evaluation and Redistribution

During optimization, it may so happen that two or more clusters get overlapped and therefore searching in a same region. So, redistribution is necessary over time. This is implemented in our algorithm with an extra technique of performance evaluation. After a time span (TS) or generations the performance of the algorithm is evaluated and accordingly the cluster numbers are updated. Performance of the algorithm is evaluated in terms of number of changes in global best value over a TS. If the change is satisfactory i.e. if number of changes is greater than a fixed value change_ref, then cluster number is reduced, otherwise it is increased. However, the number of clusters remains between two limits called max_cluster & min_cluster, which are the maximum and minimum allowed cluster numbers. After updating the number of clusters, the total population is re-clustered in updated cluster numbers. When the algorithm is performing well, it reduces the cluster number. However, when the performance evaluation test gives bad results, then the cluster number is increased by one. A fresh cluster is injected in the population, whose individuals are generated randomly. This process can help us to maintain the diversity of the population, which is an important aspect regarding to DOPs.

Self-adaptive Cluster-Based Differential Evolution

23

3.4 Detection of Environmental Change and External Archive

It was stated that an efficient algorithm should detect when the environment changes. For the detection purpose, we place a particle, called test particle, in the search space. The test particle does not take part in the optimization process; rather its objective value is constantly evaluated in each generation. If its new function value does not match with the previous value, then we can say that an environmental change has occurred. Then we stop continuing the optimization process and it is restarted afresh, i.e. the algorithm starts from the Initialization step. An algorithm can be stated efficient in dynamic environments if it can utilize the knowledge gained in the previous environment on the changed one. For this purpose, we have used an External Archive. During optimization, the radius of each cluster is calculated. If the radius becomes smaller than Rconv, described in (5), then we can say that the cluster is highly converged around a point. So then the position of the best particle of that cluster is stored in an Archive and the cluster is deleted. When a change in environment is detected, then the particles of the Archive, which were the optima in previous environment, are added to the new population, expecting they could help in tracking the movement of optima. If the change in environment is not too severe, this method should improve the search in new environment.

4 Experimental Results 4.1 Test Problems

In this paper, our algorithm is tested on the GBDG benchmark problems [2]. The test suite has 6 problems F1-F6. The number of peaks of F1 is 10, 50 and other functions have 10 peaks each. For our simulation purpose, we took the change instances T1 to T6. The performance of our algorithm is compared with other 3 significant dynamic optimizers. The functions are evaluated over 60 environment changes in a single run and 20 such runs are recorded. The mean and standard deviation values of those runs are shown in tables 1-7. 4.2 Parameter Settings

NP = 200; F = 0.5*(rand+1); Cr = 0.9; k = 5; max_cluster = 2k = 10, min_cluster = 1; pop_max = 100; TS = 10; Rconv can be calculated from (5). Table 1. Error values achieved for F1 with number of peaks = 10 Change instance Algorithm

SACDEEA

jDE CPSO DynDE

Mean Std. Mean Std. Mean Std. Mean Std.

T1

T2

T3

T4

T5

T6

1.32e-05 4.67e-05 2.88e-02 4.43e-01 3.51e-02 4.26e-01 7.32e-02 2.96e+00

1.23e+00 5.54e+00 3.59e+00 7.84e+00 2.72e+00 6.52e+00 2.55e+00 8.43e+00

1.32e+01 3.20e+01 3.00e+01 7.13e+01 4.13e+00 9.00e+00 5.42e+00 9.24e+00

9.67e-04 2.76e-03 1.53e-02 2.88e-01 9.44e-02 7.86e-01 1.26e-01 9.42e-01

7.23e-01 2.02e+00 2.18e+00 4.39e+00 1.87e+00 4.49e+00 1.56e+00 4.64e+00

1.02e+00 3.90e+00 1.15e+00 5.73e+00 1.06e+00 4.81e+00 1.31e+00 6.25e+00

24

U. Halder et al.

Table 2. Error values achieved for F1 with number of peaks = 50 Change instance

Algorithm SACD EEA

jDE CPSO DynDE

Mean Std. Mean Std. Mean Std. Mean Std.

T1 1.32e-03 3.89e-02 1.72e+00 7.64e-01 2.63e-01 9.36e-01 3.29e-01 1.52e+00

T2 1.34e+00 5.96e+00 4.09e+00 6.45e+00 3.28e+00 5.30e+00 4.65e+00 6.34e+00

T3 4.83e+00 7.29e+00 4.29e+00 6.75e+00 6.32e+00 7.44e+00 6.46e+00 9.35e+00

T4 3.87e-04 1.89e-02 8.77e-02 2.46e-01 1.25e-01 3.86e-01 1.41e-01 5.91e-01

T5 5.98e-03 4.67e-01 9.48e-01 1.77e+01 8.48e-01 1.78e+00 1.02e+00 2.65e+00

T6 5.73e-02 2.03e+00 1.77e+00 5.83e+00 1.48e+00 4.39e+00 9.86e-01 4.86e+00

T5 1.76e+01 4.32e+01 6.71e+01 1.30e+02 2.51e+01 6.43e+01 2.08e+01 6.45e+01

T6 9.83e-01 2.90e+00 3.37e+00 1.28e+01 1.99e+00 5.22e+00 2.18e+00 3.96e+00

T5 6.01e+02 2.95e+02 4.76e+02 3.80e+02 8.60e+02 1.22e+02 7.49e+02 2.81e+02

T6 2.30e+02 4.00e+02 2.43e+02 3.85e+02 7.53e+02 3.62e+02 5.19e+02 4.38e+02

Table 3. Error values achieved for F2 Change instance

Algorithm SACDEEA

jDE CPSO DynDE

Mean Std. Mean Std. Mean Std. Mean Std.

T1 3.09e-02 2.09e-01 9.63e-01 3.08e+01 1.25e+00 4.18e+00 1.36e+00 5.03e+00

T2 6.22e+00 3.12e+00 4.30e+01 1.15e+02 1.01e+01 3.51e+01 1.30e+01 4.82e+01

T3 4.54e+00 3.45e+00 5.02e+01 1.24e+02 1.03e+01 3.35e+01 1.19e+01 4.57e+01

T4 9.34e-02 3.56e-01 7.93e-01 2.53e+00 5.67e-01 2.14e+00 7.84e-01 2.22e+00

Table 4. Error values achieved for F3

Algorithm SACDEEA

jDE CPSO DynDE

Mean Std. Mean Std. Mean Std. Mean Std.

T1 1.07e+01 3.23e+01 1.14e+01 5.81e+01 1.38e+02 2.22e+02 2.12e+01 7.37e+01

T2 4.95e+02 2.04e+02 5.58e+02 3.85e+02 8.55e+02 1.61e+02 7.92e+02 2.56e+02

Change T3 5.80e+02 1.52e+02 5.72e+02 3.86e+02 7.66e+02 2.36e+02 6.36e+02 3.43e+02

instance T4 5.86e+01 1.79e+02 6.57e+01 2.09e+02 4.31e+02 4.32e+02 3.42e+02 4.19e+02

Self-adaptive Cluster-Based Differential Evolution

25

Table 5. Error values achieved for F4

Algorithm SACDEEA

jDE CPSO DynDE

Mean Std. Mean Std. Mean Std. Mean Std.

T1 1.24e+00 5.46e+00 1.49e+00 4.48e+00 2.68e+01 7.06e+00 1.86e+00 5.75e+00

T2 2.43e+00 5.67e+00 4.95e+00 1.35e+00 3.72e+01 9.94e+01 3.95e+01 9.86e+01

Change T3 1.34e+01 5.90e+01 5.19e+01 1.42e+02 3.67e+01 9.72e+01 2.35e+01 9.45e+01

instance T4 2.97e+00 2.63e+00 1.51e+00 4.10e+00 7.93e-01 2.78e+00 8.69e-01 3.17e+00

T5 1.20e+02 1.93e+01 6.94e+01 1.44e+00 6.72e+01 1.30e+02 4.47e+01 1.22e+02

T6 7.67e-01 1.09e+01 2.35e+00 5.78e+00 4.88e+00 1.54e+01 1.56e+00 6.21e+00

T5 1.38e-01 2.90e+00 4.09e-01 1.91e+00 7.99e+00 1.38e+01 8.44e+00 1.21e+01

T6 1.23e-01 3.78e-01 2.30e-01 9.35e-01 4.05e+00 8.37e+00 2.30e+00 3.62e+00

T5 1.05e+00 2.36e+01 1.49e+01 4.52e+01 7.16e+01 1.60e+02 4.34e+01 1.37e+02

T6 2.98e+00 1.13e+00 7.80e+00 1.10e+01 2.37e+01 5.16e+01 1.21e+01 2.52e+01

Table 6. Error values achieved for F5 Algorithm SACDEEA

jDE CPSO DynDE

Mean Std. Mean Std. Mean Std. Mean Std.

T1 2.90e-02 3.09e+00 1.60e-01 1.03e+00 1.86e+00 5.18e+00 2.99e+00 6.88e+00

T2 1.89e+00 2.89e+01 3.34e-01 1.64e+00 2.88e+00 6.79e+00 2.94e+00 4.72e+00

Change T3 2.74e-01 1.98e+00 3.58e-01 1.83e+00 3.40e+00 6.45e+00 2.91e+00 5.39e+00

instance T4 3.09e-02 3.02e-01 1.08e-01 8.27e-01 1.10e+00 4.87e+00 1.38e+00 2.42e+00

Table 7. Error values achieved for F6 Algorithm SACDEEA

jDE CPSO DynDE

Mean Std. Mean Std. Mean Std. Mean Std.

T1 4.78e+00 1.23e+01 6.23e+00 1.04e+01 6.73e+00 9.97e+00 6.04e+00 1.10e+01

T2 2.78e+01 1.98e+01 1.03e+01 1.32e+01 2.16e+01 6.35e+01 2.02e+01 6.22e+01

Change T3 5.98e+00 1.87e+01 1.10e+01 2.33e+01 2.71e+01 8.40e+01 1.93e+01 6.74e+01

instance T4 3.02e+00 1.78e+01 6.79e+00 1.02e+01 9.27e+00 2.42e+01 8.87e+00 2.67e+01

5 Conclusion This paper proposes a self adaptive cluster based Differential Evolution algorithm for solving DOPs. We have evaluated the performance of our algorithm on the benchmark problems of GDBG system. There are six test functions and the numerical results show satisfactory performance of the algorithm. However, we have to enhance our algorithm such that it can detect dimension change also. So, our future work will

26

U. Halder et al.

include this change instance. It will also be attempted to improve the self adaptive nature of the algorithm to get further good results.

References 1. Storn, R., Price, K.: Differential evolution – A simple and efficient heuristic for global optimization over continuous spaces. Journal of Global Optimization 11(4), 341–359 (1997) 2. Li, C., Yang, S., Nguyen, T.T., Yu, E.L., Yao, X., Jin, Y., Beyer, H.G., Suganthan, P.N.: Benchmark Generator for CEC 2009 Competition on Dynamic Optimization. University of Leicester, Univ. of Birmingham, Nanyang Technological University, Tech. Rep. (2008) 3. Grefenstette, J.J.: Genetic algorithms for changing environments. In: Proc. 2nd Int. Conf. Parallel Problem Solving from Nature, pp. 137–144 (1992) 4. Angira, R., Santosh, A.: Optimization of dynamic systems: A trigonometric differential evolution approach. Computers & Chemical Engineering 31(9), 1055–1063 (2007) 5. Mendes, R., Mohais, A.S.: DynDE: a differential evolution for dynamic optimization problems. In: Proc. of IEEE Cong. on Evol. Comput., vol. 2, pp. 2808–2815 (2005) 6. Branke, J.: Memory enhanced evolutionary algorithms for changing optimization problems. In: Proc. of IEEE Congress on Evolutionary Computation, vol. 3, pp. 1875– 1882 (1999) 7. Yang, S., Li, C.: A Clustering Particle Swarm Optimizer for Locating and Tracking Multiple Optima in Dynamic Environments. IEEE Transactions on Evolutionary Computation 14, 959–974 (2010) 8. Brest, J., Zamuda, A., Boskovic, B., Maucec, M.S., Zumer, V.: Dynamic Optimization using Self-Adaptive Differential Evolution. In: Proc. Cong. on Evol. Comput., pp. 415– 422 (2009) 9. Liu, L., Yang, S., Wang, D.: Particle Swarm Optimization with Composite Particles in Dynamic Environments. IEEE Transactions on Systems, Man, and Cybernetics—Part B: Cybernetics 40(6) (December 2010) 10. Das, S., Suganthan, P.N.: Differential Evolution: A Survey of the State-of-the-art. IEEE Trans. on Evolutionary Computation 15(1), 4–31 (2011) 11. Mallipeddi, R., Suganthan, P.N., Pan, Q.K., Tasgetiren, M.F.: Differential evolution algorithm with ensemble of parameters and mutation strategies. Applied Soft Computing 11(2), 1679–1696 (2011) 12. Neri, F., Tirronen, V.: Recent advances in differential evolution: a survey and experimental analysis. Artif. Intell., Rev. 33(1-2), 61–106 (2010) 13. Mallipeddi, R., Suganthan, P.N.: Ensemble of Constraint Handling Techniques. IEEE Trans. on Evolutionary Computation 14(4), 561–579 (2010)

An Informative Differential Evolution with Self Adaptive Re-clustering Technique Dipankar Maity, Udit Halder, and Preetam Dasgupta Dept. of Electronics and Tele-communication Engineering, Jadavpur University, Kolkata – 700032, India {dipankarmaity1991,udithalder99,dasguptapreetam}@gmail.com

Abstract. We propose an informative Differential Evolution (DE) algorithm where the information gained by the individuals of a cluster will be exchanged after a certain number of iterations called refreshing gap. The DE is empowered with a clustering technique to improve its efficiency over multimodal landscapes. During evolution, self-adaptive behaviour helps in re-clustering. With the better explorative power of the proposed algorithm we have used a new local search technique for fine tuning near a suspected optimal position. The performance of the proposed algorithm is evaluated over 25 benchmark functions and compared with existing algorithms. Keywords: Differential Evolution, optimization, cluster, self-adaptive reclustering.

1 Introduction Scientist and Engineers from all branches have to deal with the global optimization problem where the main target is to find globally maximum or minimum value for a specified objective or cost function. Traditional techniques like steepest decent, linear programming, fail to find the global optima for most of the cases. To solve these kind of problems many optimization algorithm have been invented, which are based on any natural behavior like Particle Swarm Optimization(PSO)[1] , that imitate bird flocking nature, Ant Colony Optimization (ACO)[2] is based on the foraging technique of ant, Genetic Algorithm(GA)[3] based on the principle of the Darwinian theory of the survival of the fittest and the theory of evolution of the living beings; and many others like Artificial Immune Algorithm(AIA)[4], Artificial Bee Colony(ABC) [5], Bacteria Foraging Optimization (BFO) [6], etc. Differential Evolution (DE) [7, 13-16] proposed by Storn and Price is a very simple but one of the most promising algorithm for global optimization problems and it is not motivated by biological models. Now researchers have also incorporated local search technique with GA to form Memetic Algorithm(MA), that also shows better efficiency in optimization. In many applications such as Pattern Recognition [7], Communication [8], mechanical engineering [9], Real world optimization the effectiveness and efficiency of DE and other algorithms has been successfully demonstrated. But most of the algorithms and classical DE also fails to find the global optimum (optima) when the problem is high dimensional, highly multimodal. So here we propose a new algorithm called Informative B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 27–34, 2011. © Springer-Verlag Berlin Heidelberg 2011

28

D. Maity, U. Halder, and P. Dasgupta

Differential Evolution with Self Adaptive Re-clustering Technique (IDE-SR) to find the most promising solutions for such problems. The information exchange through the re-clustering technique is the main theme of this paper. The rest of the paper is organized as follows section 2 contains related research works on the problems, section 3 is about basic DE algorithm, section 4 is the description of our proposed algorithm, section 5 gives the set of parameters and section 6 gives the experimental results of our algorithm on benchmark problem set [10] and compare with other algorithms and section 7 concludes the paper.

2 Related Research Works A lot of researches have been done for the improvement of DE and also there are a lot of research works which have tried to optimize the CEC-2005 benchmark functions. Some of them are: • A. K. Qin, V. L. Huang, and P. N. Suganthan, proposed a self-adaptive nature for solving these problems [11]. • Anne Auger and Nikolaus Hansen proposed a CMA evolution strategy by tuning the population size [12].

3 Differential Evolution (DE) Algorithm Differential Evolution (DE) is a very simple but a very powerful algorithm for optimization problem. Let S ∈ R be the search space of the problem under consideration. DE algorithm starts with an initial population of NP, n dimensional solution  particles. The particles are of the form X i = ( x1,i , x2,i , x3,i ,..., xn ,i ) ∈ S , where n

i=1,2,….,NP and are upgraded from one generation to next generation, where xi1 , xi 2 ,...., xin are in between their respective upper and lower bounds respectively. The population undergoes through Crossover, Mutation at x upper , x lower j j





each generation t and produces a new solution vector U i ,t for each vector X i,t . 3.1 Mutation





After initialization for each solution vector X i ,t , a new vector Vi ,t is generated at each



generation t. There are some methods to generate Vi ,t , DE/rand/1, DE/best/1, DE/target-to-best/1, DE/best/2, DE/rand/2 etc. are some most useful methods. We used the DE/best/1 for our algorithm.     (1) V i , t = X b e s t , t + F .( X r i , t − X r i , t ) , 1

where

2

r1i , r2i are mutually exclusive integers randomly chosen in the range [1,NP]

and both are different from the index i.

An Informative Differential Evolution with Self Adaptive Re-clustering Technique

29

3.2 Cross-Over After the phase of mutation the crossover phase plays a major role to enhance the



diversity of the population. In this phase the generated vector Vi ,t exchanges its com-



ponent with its parent vector X i ,t to generate a new vector  U i , t = ( u 1 , i ,t , u 2 ,i ,t , u 3 , i , t ,..., u n , i ,t ) , where u j ,i ,t is found by the following procedure:

 v j , i , t , if ra nd i , j (0,1) < C r u j , i ,t =   x j , i , t , o th erw ise where rand

i, j

j = j ra n d

or

( 0 ,1 ) is a uniformly distributed random number in the range (0,1).



jrand ∈ [1,2,...., n] is a randomly chosen index, which ensures that U i , t gets at least  one component from Vi ,t , and Cr is a user defined constant in the range [0,1]. 3.3 Selection Operation In this phase a greedy selection is made between the target vector and generated vector for next generation. The selection procedure is done by the following way:

 X i ,t +1

 U i ,t =   X i ,t

  if f (U i ,t ) ≤ f ( X i ,t )   if f (U i ,t ) > f ( X i ,t )

for minimization problem. Thus after every generation we find either a new solution which has better fitness (here for minimization problem) or the previous vector is kept. So after each generation the population gets better or remains unchanged but never deteriorates.

4 Proposed Algorithm In our proposed algorithm we have used an information exchange strategy. We have adopted a self-adaptive cluster based strategy with a novel Local Search technique. The total population is divided into some clusters using K-means clustering algorithm; this is done to avoid the trapping of particles in any local optimum i.e. if the population is not divided then they can prematurely converge to a single local optima, whereas if they are clustered then it is expected that even though the premature convergence occurs, but different cluster will be converged to different optima. Each cluster exchanges information only after a certain number of iterations. After those iterations - which will be called as refreshing gap (g_r) for the rest of the paper- the total population is again clustered, now the cluster number may change or remain same. By this re-clustering technique, particles that were in different clusters in previous g_r can belong to same cluster now. So there will be a chance among

30

D. Maity, U. Halder, and P. Dasgupta

those particles to exchange information that they have learned in past g_r. This information exchange helps them to explore a better position than that could be found by not using this re-clustering technique. The cluster number is varied in a self adaptive manner, here we used that if the algorithm performs satisfactorily well then cluster number is decreased until the cluster number is cluster_min, otherwise it is increased until it is cluster_max. It may so happen that two or more clusters can come closer to each other while improving them, then in the next g_r when we will re-cluster the total population then these clusters will form a single group as the K-means algorithm generates clusters based on the spatial distribution of the particles. So this algorithm does not require further checking of the overlapping of two or more clusters. We used DE/best/1 to each cluster separately because this variation of DE has faster convergence. In multimodal landscapes when a cluster has converged to an optimum then all the particles in that cluster come very closer so the difference vector in equation (1) becomes very small. Due to this premature convergence the rest of the FEs are wasted. So to save FE and for the fine tuning of the suspected optimum we have used a Local Search technique. In each iteration we calculated the radius of the clusters by the following rule, m

R =

  (x i =1

i

 T   − c ) .( x i − c ) ,

m

(2)



where m is the number of particles and c is the centre of the cluster defined as m

 c =



i =1

 xi .

(3)

m

If R is less than a pre-specified value R_limit, then we use the Local Search technique for better exploration in the small range, otherwise we use the DE/best/1.

R _ limit =

(X

upper

)(

)

   T − X lower . X upper − X lower .10 −2 .

(4)

A brief description of the algorithm is as follows: 4.1 Initialization We initialize the population randomly in the search space that try to cover the whole search region uniformly as much as possible. Using K-means algorithm we divide the population into clusters with cluster number cluster_max. 4.2 Information Exchange among Clusters For each g_r each particle is updated using DE/best/1 and they can exchange information among the particles that belong to that cluster only. If the radius of the cluster becomes less than R_limit then we use the local search algorithm in search of better

An Informative Differential Evolution with Self Adaptive Re-clustering Technique

31

solution within that region. If any cluster has radius less than R_limit and does not improve after applying the Local Search technique then the cluster is deleted unless the cluster contains the global best particle. We calculate the percentage change of the function value after each g_r and accordingly cluster number is varied. If percentage change is less than p_ref then cluster_no = max( cluster_min, prev_cluster_no-1). Otherwise

cluster_no = min( cluster_max, prev_cluster_no+1).

4.3 Local Search Technique We generate some exploiting particles (exploiters) to exploit information from the suspected optimum position. When a cluster radius is less than R_limit then almost all the particles in that cluster are very close to each other, so if we do Local Search around each particle then it will be some sort of wastage of FEs. So we take only best 40% particles of that cluster and new particles are randomly generated around these 40% of old particles within a radius r_exploit. The r_exploit is small for the best fit particle and comparatively large for the worst particle among those 40% of particles. The number of exploiters generated is varied according to the fitness of the particles. We used 2     (exploiters _max − exploiters _min )1 −  f i − best      worst − best    , exploiters i = round       + exploiters _min  

(5)

Where fi is the fitness value of that particle and best and worst are the fitness value of the best fit and worst fit particle respectively. In case of r_exploit we used

(

( f − best ) ) worst + r _ ini , − best

r _ exploit = 1 − k d .

i

Where

k=

best . worst

(6)

5 Parameters NP = 150, cluster_min = 2, cluster_max = 10 , exploitrs_max = 7, exploiters_min = 2, d = 1, g_r = 10, F = 0.5( rand(0,1) + 1) , Cr =0.9, r_ini = 5e-03, p_ref = 5.

6 Experimental Results We have applied our algorithm on the 25 benchmark problem of CEC-2005 and the dimensionalities of those functions are 10 and 30. We have run the algorithm 25 times on each function and calculated the mean and standard deviation (std) from that results.

32

D. Maity, U. Halder, and P. Dasgupta

Table 1. Error values for dimension n = 10 Func1

Func2

Func3

Func4

Func5

Func6

Func7

Mean Std

Mean Std

Mean Std

Mean Std

Mean Std

Mean Std

Mean Std

5.14e-09 1.82e-09 0.00e+00 0.00e+00 0.00e+00 0.00e+00 Func8

5.31e-09 1.77e-09 1.04e-13 5.11e-13 0.00e+00 0.00e+00 Func9

4.94e-09 1.45e-09 1.67e-05 3.12e-05 0.00e+00 0.00e+00 Func10

1.79e+06 4.66e+06 1.41e-05 7.09e-05 0.00e+00 0.00e+00 Func11

6.57e-09 1.88e-09 1.2e-02 1.4e-02 3.63e-12 1.99e-12 Func12

5.41e-09 1.81e-09 1.20e-08 1.93e-08 0.00e+00 0.00e+00 Func13

4.91e-09 1.68e-09 1.99e-02 1.07e-02 4.70e-02 2.25e-04 Func14

SADE

Mean Std 2.00e+01 0.00e+00 2.00e+01 5.39e-08

Mean Std 4.49e+01 1.36e+01 0.00e+00 0.00e+00

Mean Std 4.08e+01 3.35e+01 4.96e+00 1.69e+00

Mean Std 3.65e+00 1.66e+00 4.89e+00 6.62e-01

Mean Std 2.09e+02 4.69e+02 4.50e-07 8.50e-07

Mean Std 4.94e-01 1.38e-01 2.20e-01 4.00e-02

Mean Std 4.01e+01 3.14e-01 2.91e+00 2.06e-01

IDE-SR

2.00e+01 0.00e+00

3.75e+00 1.98e+00

3.70e+00 1.38e-01

1.17e+00 3.99e-01

1.02e-12 6.35e-11

2.10e-01 1.90e-03

2.46e+00 1.50e-01

Func15

Func16

Func17

Func18

Func19

Func20

Func21

Mean Std 2.11e+02 1.02e+02 3.20e+01 1.10e+02 8.65e+01 1.64e+01

Mean Std 1.05e+02 1.26e+01 1.01e+02 6.17e+00 8.03e+01 3.56e+01

Mean Std 5.49e+02 3.49e+02 1.14e+02 9.97e+00 9.17e+01 2.79e+01

Mean Std 4.97e+02 2.18e+02 7.19e+02 2.08e+02 3.00e+02 1.24e-01

Mean Std 5.16e+02 2.34e+02 7.05e+02 1.90e+02 3.43e+02 2.47e+01

Mean Std 4.42e+02 2.03e+02 7.13e+02 2.01e+02 3.17e+02 1.67e+01

Mean Std 4.04e+02 1.23e+02 4.64e+02 1.58e+02 3.01e+02 8.61e+00

Func22

Func23

Func24

Func25

Mean Std

Mean Std

Mean Std

Mean Std

7.40e+02 5.94e+01 7.35e+02 9.15e+01 7.11e+02 2.58e+01

7.91e+02 2.79e+02 6.64e+02 1.53e+02 5.58e+02 6.46e-01

8.65e+02 6.39e+02 2.00e+02 0.00e+00 2.98e+02 1.84e+01

4.42e+02 3.58e+02 3.76e+02 3.14e+00 3.11e+02 1.85e+01

Algorithm GCMA SADE IDE-SR Algorithm GCMA

Algorithm GCMA SADE IDE-SR Algorithm GCMA SADE IDE-SR

An Informative Differential Evolution with Self Adaptive Re-clustering Technique

33

Table 2. Error values for dimension n = 30 Func1

Func2

Func3

Func4

Func5

Func6

Func7

Mean Std

Mean Std

Mean Std

Mean Std

Mean Std

Mean Std

Mean Std

GCMA

5.42e-09 9.80e-10

6.22e-09 8.95e-10

5.55e-09 1.09e-09

1.11e+04 3.02e+04

8.62e-09 8.53e-10

5.90e-09 1.61e-09

5.31e-09 1.41e-09

SADE

0.00e+00 0.00e+00

9.72e-08 4.86e-07

5.05e+04 1.58e+05

5.82e-06 1.45e-05

7.88e+02 1.24e+03

2.12e+01 1.34e+01

8.27e-03 1.14e-02

IDE-SR

0.00e+00 0.00e+00

1.64e-13 6.69e-14

6.38e+02 1.59e+02

2.36e-02 5.34e-02

6.21e+02 4.05e+01

1.17e+00 1.45e+00

5.21e-06 1.57e-05

Func8

Func9

Func10

Func11

Func12

Func13

Func14

Mean Std 2.01e+01 2.79e-01

Mean Std 9.38e-01 1.18e+00

Mean Std 1.65e+00 1.35e+00

Mean Std 5.48e+00 3.13e+00

Mean Std 4.43e+04 2.19e+05

Mean Std 2.49e+00 5.13e-01

Mean Std 1.29e+01 4.19e+01

SADE

2.01e+01 5.73e-03

2.27e-15 1.14e-14

3.58e+01 6.08e+00

2.66e+01 1.13e+00

8.73e+02 9.34e+02

1.21e+00 1.34e-01

1.24e+01 2.59e-01

IDE-SR

2.01e+01 1.35e-05

1.41e-02 1.84e-02

1.12e+00 5.21e-02

4.99e+00 1.25e+00

1.32e+02 9.71e+01

1.04e+00 7.39e-01

1.14e+01 3.44e-01

Func15

Func16

Func17

Func18

Func19

Func20

Func21

Mean Std 2.08e+02 2.75e+01 3.28e+02 9.65e+01

Mean Std 3.50e+01 2.04e+01 1.38e+02 1.70e+01

Mean Std 2.91e+02 1.93e+02 1.51e+03 9.63e+02

Mean Std 9.04e+02 2.88e-01 9.54e+02 3.44e+01

Mean Std 9.04e+02 2.71e-01 8.46e+02 6.22e+02

Mean Std 9.04e+02 2.48e-01 2.02e+03 8.77e+02

Mean Std 5.00e+02 1.31e-13 1.73e+03 5.12e+02

1.15e+02 2.69e+00

7.51e+01 1.53e+01

9.42e+01 1.17e+01

8.49e+02 3.86e+00

8.46e+02 3.53e+00

8.30e+02 3.92e+00

5.00e+02 3.26e-15

Func22

Func23

Func24

Func25

Mean Std

Mean Std

Mean Std

Mean Std

GCMA

8.03e+02 1.86e+01

5.34e+02 2.22e-04

9.10e+02 1.48e+02

2.11e+02 9.21e-01

SADE

1.58e+03 4.52e+02

5.21e+02 2.49e+01

2.00e+02 1.25e-02

5.00e+02 5.68e-03

IDE-SR

5.45e+02 4.43e+00

4.98e+02 1.72e+01

2.00e+02 5.61e-03

2.07e+02 1.15e+00

Algorithm

Algorithm GCMA

Algorithm GCMA SADE IDE-SR

Algorithm

7 Conclusion In this paper, we have proposed an Informative Differential Evolution with Self Adaptive Re-clustering Technique for solving global optimization problems. The cluster number is changed in a self adaptive way. The proposed Local Search technique also takes the advantage of self-adaptive behaviour of the algorithm to change

34

D. Maity, U. Halder, and P. Dasgupta

the R_limit parameter which is crucial for the performance of the Local Search algorithm and thus for the performance of IDE-SR.

References 1. Kennedy, J., Eberhart, R.C.: Particle Swarm Optimization. In: Proceedings of IEEE International Conference on Neural Networks, Piscataway, NJ, pp. 1942–1948 (1995) 2. Dorigo, M., Stützle, T.: Ant Colony Optimization. MIT Press, Cambridge (2004) 3. Holland, J.: Adaptation in Natural and Artificial Systems. University of Michigan Press, Ann Arbor (1975) 4. Farmer, J.D., Packard, N., Perelson, A.: The Immune System, Adaptation and Machine Learning. Physica D 22, 187–204 (1986) 5. Karaboga, D.: An Idea Based on Honey Bee Swarm for Numerical Optimization, technical REPORT-TR06, Erciyes University, Engineering Faculty, Computer Engineering Department (2005) 6. Passino, K.M.: Biomimicry of Bacterial Foraging for Distributed Optimization and Control. IEEE Control Systems Magazine 22, 52–67 (2002) 7. Storn, R., Price, K.V.: Differential Evolution-A simple and efficient Heuristic for Global Optimization over continuous Spaces. Journal of Global Optimization 11, 341–359 (1997) 8. Ilonen, J., Kamarainen, J.K., Lampinen, J.: Differential Evolution Training Algorithm for Feed- Forward Neural Networks. Neural Processing Letters 7, 93–105 (2003) 9. Storn, R.: Differential evolution design of an IIR-filter. In: Proceedings of IEEE Int. Conference on Evolutionary Computation, ICEC 1996, pp. 268–273. IEEE Press, New York (1996) 10. Suganthan, P.N., Hansen, N., Liang, J.J., Deb, K., Chen, Y.-P., Auger, A., Tiwari, S.: Problem Definitions and Evaluation Criteria for the CEC 2005 Special Session on RealParameter Optimization. Nanyang Technological University, Tech. Rep. (2005) 11. Qin, A.K., Huang, V.L., Suganthan, P.N.: Differential evolution algorithm with strategy adaptation for global numerical optimization. IEEE Trans. on Evolutionary Computations, 398–417 (April 2009), doi:10.1109/TEVC.2008.927706 12. Auger, A., Hansen, N.: A Restart CMA Evolution Strategy With Increasing Population Size. In: Proceedings of the IEEE Congress on Evolutionary Computation, Piscataway, NJ, USA, vol. 2, pp. 1769–1776. IEEE Press (2005) 13. Das, S., Suganthan, P.N.: Differential Evolution: A Survey of the State-of-the-art. IEEE Trans. on Evolutionary Computation 15(1), 4–31 (2011) 14. Mallipeddi, R., Suganthan, P.N., Pan, Q.K., Tasgetiren, M.F.: Differential evolution algorithm with ensemble of parameters and mutation strategies. Applied Soft Computing 11(2), 1679–1696 (2011) 15. Ghosh, A., Das, S., Chowdhury, A., Giri, R.: An improved differential evolution algorithm with fitness-based adaptation of the control parameters. Inf. Sci. 181(18), 3749–3765 (2011) 16. Das, S., Abraham, A., Chakraborty, U.K., Konar, A.: Differential Evolution Using a Neighborhood-Based Mutation Operator. IEEE Transactions on Evolutionary Computation 13(3), 526–553 (2009)

Differential Evolution for Optimizing the Hybrid Filter Combination in Image Edge Enhancement Tirimula Rao Benala1, Satchidananda Dehuri2, G.S. Surya Vamsi Sirisetti1, and Aditya Pagadala1 1

Anil Neerukonda Institute of Technology and Sciences Sangivalasa, Visakhapatnam, Andhra Pradesh, India [email protected], [email protected], [email protected] 2 Department of Information & Communication Technology Fakir Mohan University, Vyasa Vihar, Balasore-756019, India [email protected]

Abstract. Image edge enhancement is the art of enhancing the edge of significant objects in an image. The proposed work uses the concept of hybrid filters for edge enhancement whose optimal sequence is to be found by differential evolution. Its unbiased stochastic sampling and bench-mark results in a quite many number of applications ignited us to use for the aforesaid purpose and motivated for further research. The major five mutational variants of differential evolution employing the binomial crossover have been used in the proposed work which and have been tested over both standard images and medical images. Our extensive experimental studies produce encouraging results. Keywords: Image edge enhancement, hybridized smoothening filter, differential evolution.

1 Introduction In the modern era, digital images have been widely used in an aggrandizing number of applications and the effort on edge enhancement has been focused mostly to improve visual perception of images that are unclear (blurred). Edges are the representations of the discontinuities of image intensity functions. For processing these discontinuities in an image, an efficient edge enhancement technique is essential [5]. These edge enhancement techniques fall under two categories: smoothening filters and sharpening filters [2]. Smoothening filters are used for blurring and noise reduction [6]. Noise reduction can be accomplished by blurring with linear filters (mean, median and mode) and non-linear filters (circular, pyramidal and cone) [8]. Sharpening filters (Laplacian, Sobel, Prewitt and Robert filters) are used to highlight fine details in an image or to enhance details that have been blurred but because of their results of complexity and image quality, smoothening filters are used which involves simple subtractive smoothened image concept which reduces complexity and makes the images look sharper than they really are. This can also be done with the help of a new filter called B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 35–42, 2011. © Springer-Verlag Berlin Heidelberg 2011

36

T.R. Benala et al.

the hybridized smoothening filter (a sequence of smoothening filter) [8]. The optimal magnitude (for different combinations of smoothening filters) of the hybrid filter is found by using the differential evolution (DE) and is compared to that obtained by the different variants of it. Hybrid filters, its optimization using DE is explained in Section 2. Section 3 discusses our proposed framework. In Section 4, experiments have been conducted and their corresponding results obtained are discussed. Conclusions and further study is given in Section 5.

2 Hybrid Filters and Differential Evolution Algorithm 2.1 Hybrid Filters Hybrid filter is defined as the series of existing filters (Smoothening filters) to optimize the magnitude of the image [8]. It can efficiently remove large amounts of mixed Gaussian and impulsive noise besides preserving the image details. In this approach, hybrid filter is taken as a combination of smoothening filters (for e.g. 1-23-4-5-6 i.e. suppose 1-mean, 2-median, 3-mode, 4-circular, 5-pyramidal, 6-cone. The output of mean filter is taken as input for median filter and the output of median filter is given as input to the next and so on). This hybrid filter yields optimal threshold values using clustering algorithm. 2.2 Differential Evolution DE is an evolutionary optimization technique, which is simple, significantly faster and robust at numerical optimization and likely chances of finding true global optimum. The stages involved are: (i) (ii)

Initialization. The DE starts with the creation of an initial population, normally at random. Mutation. New individuals are created applying the mutation operations. At each generation, for each individual of the population a new solution (v i t+1 ) is created using a weighted vector difference between two other individuals, selected randomly from the current population. The available schemes use the notation DE/α/γ/β where α represents the way in which individuals are selected from the current population. They can be selected either randomly (rand) or as the best individual from the current population (best). The γ is the value of difference vector pairs used, which normally is 1 or 2. t represents the tth generation and r1, r2, etc. are randomly selected individuals from the tth generation population. F value is the scaling factor, which controls the differential variation (mutation). The different types of mutations are listed in table1. v i t+1 = xr3t + F(x r1t- x r2t)

(1)

(iii) Crossover. The crossover which has been applied here is of binomial type which follows the scheme given in equation 2. For each individual x i t, a trial individual u i t+1 is generated by using equation 2 which involves using a crossover constant cr [10]

Differential Evolution for Optimizing the Hybrid Filter Combination

u i t+1 =

v i t+1 if (rand < cr) || j==rand(0,D) xi

t

37

(2)

else if(rand> cr)

Table 1. Mutation Schemes [4]

Scheme name DE/ran/1/β DE/best/1/ β DE/ran/2/ β DE/best/2/ β DE/rantobest/1/ β

Mutation Definition v i t+1= xr3t+F(x r1t- x r2t) v i t+1= x best t +F(x r1t- x r2t) t+1 V i = xr5t+F(x r1t- x r2t+ x r3t- x r4t) v i t+1= x best t +F(x r1t- x r2t+ x r3t- x r4t) v i t+1= xr3t+ F(x best t- x r3t) +F(x r1t- x r2t)

(iv) Selection. The child individual x (i, t+1) is selected between each pair of x (i, t) and u (i, t+1) by using greedy selection criterion: x i t+1 =

u i t+1 xi

t

if F(u i t+1) < F(x i t)

(3)

other wise,

where, F is the fitness function.

3 Framework 3.1 Initialization of Population First optimal magnitudes of the image are obtained by using adaptive thresh-holding with foreground and background clustering algorithm [9] as described below: One background and one foreground are assumed, K=2, i.e. only two clusters are considered, which makes the overall implementation easy. (i) Region selection. Divide the document into all-inclusive mutually exclusive sub-regions. Select the document sub-region for which the threshold be computed, and a region containing the sub-region that will be used to determine the threshold sub-region. For example, the region may consist of N contiguous scan lines, where the sub-region is the center M scan lines, with M 1 hour > 1 day

Different threshold valu ues of test images, obtained by using meta-heuristics, are given in Table 3. Experimeental results of exhaustive search is avoided because off its times complexity. For con nvenience, best results are shown in bold letters. For the performance evolution of DE, PSO and GA correlation, PSNR, Mean Structtural ment (MSSIM) [14] and Universal Image Quality Inndex Similarity Index Measurem (UIQI) [15] between original image and segmented image are measured. Correlation and PSNR are tabulated in Table 4, whereas MSSIM and UIQI are shown in Table 5. It showss that results are almost similar for lower no. of threshhold values, but for higher no thrreshold DE has an advantage in the most of the cases.

56

S. Sarkar, G.R. Patra, and S. Das

(a)

(a')

(a'')

(b)

(b')

(b'')

(c)

(c')

(c'')

(d)

(d')

(d'')

Fig. 2. Thresholded images obtained by MCET-DE method [(a), (b), (c), (d) 2-level thresholding, (a'), (b'), (c'), (d') 3-level thresholding, (a''), (b''), (c''), (d'') 4-level thresholding)] Table 2. Mean objective function value ( L 2 3 4

DE -4.012e+007 -4.016e+007 -4.017e+007

) and standard deviation ( PSO

0.0 0.0 0.0

-4.008e+007 -4.013e+007 -4.013e+007

3.496e+004 3.816e+004 4.207e+004

) of “Lena” GA

-4.011e+007 3.617e+003 -4.016e+007 21.070e+000 -4.017e+007 1.515e+000

A Differential Evolution Based Approach for Multilevel Image Segmentation

57

Table 3. Threshold values acquired by using DE, GA, PSO Image Lena

Baboon

Boat

Cameraman

L

DE

2 3 4 2 3 4 2 3 4 2 3 4

84,142 75,120,166 73, 110, 142, 177 89,144 71,111,153 59, 95, 128, 162 69 ,132 58, 113, 163 48, 88, 130, 166 51,137 31, 84, 144 31, 79, 131, 162

Threshold values PSO 83,142 77,122,166 74, 111,137, 176 88,143 71, 110, 152 56, 96,126, 162 69 ,132 57, 110, 160 48, 94, 134, 166 51,134 29,84,137 29, 69, 122, 158

GA 83,141 75, 120, 165 62,94, 130, 170 89,144 70 , 110, 152 61, 96, 128,161 64, 128 59, 114, 162 43, 82, 127, 166 49, 137 30, 83, 144 29, 76, 125, 158

Table 4. Correlation and PSNR of test images Image Lena

Baboon

Boat

Cameraman

L 2 3 4 2 3 4 2 3 4 2 3 4

DE 0.9204 0.9480 0.9605 0.9230 0.9429 0.9466 0.9124 0.9325 0.9434 0.9610 0.9657 0.9584

Correlation PSO 0.9200 0.9483 0.9612 0.9223 0.9429 0.9463 0.9124 0.9275 0.9404 0.9605 0.9637 0.9572

GA 0.9197 0.9478 0.9555 0.9229 0.9424 0.9475 0.9107 0.9357 0.9399 0.9605 0.9657 0.9612

DE 11.9622 15.5203 17.0688 12.0785 13.7640 15.3353 9.7301 15.8764 17.0920 11.3301 11.9115 14.2777

PSNR(dB) PSO 11.9736 15.4761 16.8667 12.0231 13.6677 15.3270 9.7301 15.4506 17.1636 11.1359 11.4522 13.5395

GA 11.8788 15.4566 16.7030 12.0785 13.6862 15.1896 9.4186 15.7879 17.0427 11.3207 11.9091 13.5476

Table 5. MSSIM and UIQI of test images Image Lena

Baboon

Boat

Cameraman

No. of Levels 2 3 4 2 3 4 2 3 4 2 3 4

DE 0.5181 0.6144 0.6489 0.5435 0.6436 0.7199 0.4660 0.5760 0.6396 0.5840 0.6138 0.6159

MSSIM PSO 0.5201 0.6085 0.6487 0.5429 0.6406 0.7194 0.4660 0.5590 0.6435 0.5802 0.6058 0.6403

GA 0.5210 0.6122 0.6609 0.5435 0.6423 0.7143 0.4754 0.5724 0.6453 0.5843 0.6139 0.6412

DE 0.7550 0.8645 0.9032 0.7294 0.7997 0.8450 0.6921 0.8590 0.8883 0.8186 0.8369 0.8863

UIQI PSO 0.7551 0.8642 0.9005 0.7278 0.7971 0.8456 0.6921 0.8472 0.8843 0.8127 0.8230 0.8743

GA 0.7520 0.8631 0.8911 0.7294 0.7981 0.8407 0.6863 0.8563 0.8870 0.8178 0.8366 0.8734

58

6

S. Sarkar, G.R. Patra, and S. Das

Conclusion

The objective of image thresholding for image segmentation is to obtain better segmentation effect within a short span of time. In this paper we have proposed a scheme based on differential evolution for multiple thresholding using MCET. This technique was applied to various real images and the results demonstrated the efficiency and feasibility. The results are encouraging for further research on complex image segmentation and recognition problems. New variations of DE also can be explored in future to achieve improved performance.

References [1] Sezgin, M., Sankur, B.: Survey over image thresholding techniques and quantitative performance evaluation. Journal of Electronic Imaging 13(1), 146–165 (2004) [2] Sahoo, P.K., Wilkins, C., Yeager, J.: Threshold selection using Renyi’s entropy. Pattern Recognition 30, 71–84 (1997) [3] Albuquerque, M., Esquef, I.A., Mello, A.R.G., Albuquerque, M.: Image thresholding using Tsallis entropy. Pattern Recognition Letters 25, 1059–1065 (2004) [4] Li, C.H., Tam, P.K.S.: An iterative algorithm for minimum cross entropy thresholding. Pattern Recognition Letters 19, 771–776 (1998) [5] Yin, P.: Multilevel minimum cross entropy threshold selection based on particle swarm optimization. Applied Mathematics and Computation 184, 503–513 (2007) [6] Tanga, K., Sun, T., Yang, J., Gao, S.: An improved scheme for minimum cross entropy threshold selection based on genetic algorithm. Knowledge-Based Systems (2011) [7] Hammouche, K., Diaf, M., Siarry, P.: A comparative study of various meta-heuristic techniques applied to the multilevel thresholding problem. Engineering Applications of Artificial Intelligence 23, 676–688 (2010) [8] Storn, R., Price, K.V.: Differential Evolution - a simple and efficient adaptive scheme for global optimization over continuous spaces, Technical Report TR-95-012, ICSI (1995), http://http.icsi.berkeley.edu/~storn/litera.html [9] Das, S., Suganthan, P.N.: Differential evolution – a survey of the state-of-the-art. IEEE Transactions on Evolutionary Computation 15(1), 4–31 (2011) [10] Qu, B.Y., Suganthan, P.N.: Multi-Objective Evolutionary Algorithms based on the Summation of Normalized Objectives and Diversified Selection. Information Sciences 180(17), 3170–3181 (2010) [11] Pal, S., Qu, B.Y., Das, S., Suganthan, P.N.: Linear Antenna Arrays Synthesis with Constrained Multi-objective Differential Evolution. In: Progress in Electromagnetics Research, PIER B, vol. 21, pp. 87–111 (2010) [12] Kullback, S.: Information theory and statistics. Dover, New York (1968) [13] Pal, N.R.: On Minimum Cross-Entropy Thresholding. Pattern Recognition 29(4), 575– 580 (1996) [14] Wang, Z., Bovik, A.C., Sheikh, H.R., Simoncelli, E.P.: Image Quality Assessment: From Error Visibility to Structural Similarity. IEEE Transactions on Image Processing 13(4) (2004) [15] Wang, Z., Bovik, A.C.: A Universal Image Quality Index. IEEE Signal Processing Letters 9(3), 81–84 (2002)

Tuning of Power System Stabilizer Employing Differential Evolution Optimization Algorithm Subhransu Sekhar Tripathi and Sidhartha Panda Department of Electrical and Electronics Engineering, National Institute of Science and Technology (NIST), Berhampur, Orissa 761 008, India [email protected], [email protected]

Abstract. In this paper, differential evolution (DE) optimization algorithm is applied to design robust power system stabilizer (PSS). The design problem of the proposed controller is formulated as an optimization problem and DE is employed to search for optimal controller parameters. By minimizing the timedomain based objective function, in which the deviation in the oscillatory rotor speed of the generator is involved; stability performance of the system is improved. The non-linear simulation results are presented under wide range of operating conditions; disturbances at different locations as well as for various fault clearing sequences to show the effectiveness and robustness of the proposed controller and their ability to provide efficient damping of low frequency oscillations. Keywords: Differential evolution, power system stabilizer, low frequency oscillations, power system stability.

1 Introduction Low frequency oscillations are observed when large power systems are interconnected by relatively weak tie lines. These oscillations may sustain and grow to cause system separation if no adequate damping is available [1]. Power system stabilizers (PSS) are now routinely used in the industry to damp out oscillations. An appropriate selection of PSS parameters results in satisfactory performance during system disturbances [2]. The problem of PSS parameter tuning is a complex exercise. A number of conventional techniques have been reported in the literature pertaining to design problems of conventional power system stabilizers namely: the eigenvalue assignment, mathematical programming, gradient procedure for optimization and also the modern control theory [3]. Unfortunately, the conventional techniques are time consuming as they are iterative and require heavy computation burden and slow convergence. In addition, the search process is susceptible to be trapped in local minima and the solution obtained may not be optimal [4-7]. Differential Evolution (DE) is a branch of evolutionary algorithms developed by Rainer Stron and Kenneth Price in 1995 [8] is an improved version of GA for faster optimization. DE is a population based direct search algorithm for global optimization capable of handling nondifferentiable, nonlinear and multi-modal objective functions, with few, easily chosen, control parameters. The major advantages of DE are its B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 59–67, 2011. © Springer-Verlag Berlin Heidelberg 2011

60

S.S. Tripathy and S. Panda

simple structure, ease of implementation and robustness. DE differs from other Evolutionary Algorithms (EA) in the mutation and recombination phases. DE uses weighted differences between solution vectors to change the population whereas in other stochastic techniques such as Genetic Algorithm (GA) and Expert Systems (ES), perturbation occurs in accordance with a random quantity. DE employs a greedy selection process with inherent elitist features. Also it has a minimum number of EA control parameters, which can be tuned effectively [9-10]. It has been reported in the literature that DE is far more efficient and robust compared to PSO and the EA [11]. In view of these advantages, a DE-based approach for design of power system stabilizer is proposed in this paper.

2 System Under Study The SMIB power system with SSSC controller, as shown in Fig. 1, is considered in this study. The system comprises a generator connected to an infinite bus through a step-up transformer and a SSSC followed by a double circuit transmission line. In the figure T represents the transformer; VT and VB are the generator terminal and infinite bus voltage respectively; I is the line current and PL is the real power flow in the transmission lines. The generator is equipped with hydraulic turbine & governor (HTG), excitation system and a power system stabilizer.

T

I Tr. Line

PL Bus1 Generator

Bus2

PL1

Infinite-bus

Fig. 1. Single-machine infinite-bus power system

3 The Proposed Approach 3.1 Structure of the Power System Stabilizer The structure of PSS, to modulate the excitation voltage is shown in Fig. 2. The structure consists of a sensor, a gain block with gain KP, a signal washout block and two-stage phase compensation blocks as shown in Fig. 2. The input signal of the proposed controller is the speed deviation (∆ω), and the output is the stabilizing signal VS which is added to the reference excitation system voltage. The signal washout block serves as a high-pass filter, with the time constant TW, high enough to allow signals associated with oscillations in input signal to pass unchanged. From the viewpoint of the washout function, the value of TW is not critical and may be in the range of 1 to 20 seconds [1]. The phase compensation block (time constants T1, T2 and T3, T4) provides the appropriate phase lead characteristics to compensate for the phase lag between input and the output signals.

Tuning of PSS Employing Differential Evolution Optimization Algorithm

Δω

KP

Input Sensor

Gain block

sTW 1 + sTW

Washout block

1 + sT1 1 + sT2

1 + sT3 1 + sT4

Two-stage lead-lag block

VSmax

61

VS Output

VSmin

Fig. 2. Structure of power system stabilizer

3.2 Problem Formulation \In case of lead-lag structured PSS, the sensor and the washout time constants are usually specified. In the present study, a sensor time constant TSN = 15 ms and washout time constant TW =10s are used. The controller gain KP and the time constants T1, T2, T3 and T4 are to be determined. In the present study, an integral time absolute error of the speed deviations is taken as the objective function expressed as follows: t = t sim

J =  [ | Δω | ] ⋅ t.dt

(1)

t =0

In the above equations, Δω denotes the rotor speed deviation for a set of controller parameters (note that here the controller parameters represent the parameters to be optimized; KP, T1, T2, T3 and T4; the parameters of the PSS), and tsim is the time range of the simulation. For objective function calculation, the time-domain simulation of the power system model is carried out for the simulation period. It is aimed to minimize this objective function in order to improve the system response in terms of the settling time and overshoots.

4 Overview of Differential Evolution Optimization Algorithm Differential Evolution (DE) algorithm is a stochastic, population-based optimization algorithm introduced by Storn and Price in 1996 [8]. DE works with two populations; old generation and new generation of the same population. The size of the population is adjusted by the parameter NP. The population consists of real valued vectors with dimension D that equals the number of design parameters/control variables. The population is randomly initialized within the initial parameter bounds. The optimization process is conducted by means of three main operations: mutation, crossover and selection. In each generation, individuals of the current population become target vectors. For each target vector, the mutation operation produces a mutant vector, by adding the weighted difference between two randomly chosen vectors to a third vector. The crossover operation generates a new vector, called trial vector, by mixing the parameters of the mutant vector with those of the target vector. If the trial vector obtains a better fitness value than the target vector, then the trial vector replaces the target vector in the next generation. The evolutionary operators are described below [3, 5];

62

S.S. Tripathy and S. Panda

4.1 Inilization

Chromosome For each parameter j with lower bound X Lj and upper bound X Uj , initial parameter values are usually randomly selected uniformly in the interval [ X Lj , X Uj ]. 4.2 Mutation

For a given parameter vector X i ,G , three vectors ( X r1,G X r 2,G X r 3,G ) are randomly

V selected such that the indices i, r1, r2 and r3 are distinct. A donor vector i ,G +1 is created by adding the weighted difference between the two vectors to the third vector as: Vi ,G +1 = X r1,G + F .( X r 2,G − X r 3,G )

(2)

Where F is a constant from (0, 2). 4.3 Crossover

The Three parents are selected for crossover and the child is a perturbation of one of them. The trial vector U i ,G +1 is developed from the elements of the target vector ( X i ,G ) and the elements of the donor vector ( X i ,G ). Elements of the donor vector enters the trial vector with probability CR as:

V j , i , G +1 if rand j ,i ≤ CR or j = I rand U j , i , G +1 =   X j , i , G +1 if rand j ,i > CR or j ≠ I rand

(3)

With rand j , i ~ U (0,1), Irand is a random integer from (1,2,….D) where D is the solution’s dimension i.e number of control variables. Irand ensures that Vi , G +1 ≠ X i ,G . 4.4 Selection

An The target vector X i ,G is compared with the trial vector Vi , G +1 and the one with the better fitness value is admitted to the next generation. The selection operation in DE can be represented by the following equation:

U i,G +1 if f (U i,G +1 ) < f ( X i,G ) X i,G +1 =  otherwise.  X i ,G where i ∈ [1, N P ] .

(4)

Tuning of PSS Employing Differential Evolution Optimization Algorithm

63

5 Results and Discussions The SimPowerSystems (SPS) toolbox is used for all simulations and SSSC-based damping controller design [12]. SPS is a MATLAB-based modern design tool that allows scientists and engineers to rapidly and easily build models to simulate power systems using Simulink environment. In order to optimally tune the parameters of the PSS, as well as to assess its performance, the model of example power system shown in Fig. 1 is developed using SPS blockset. 5.1

Application of DE

For the purpose of optimization of Eq. (4), DE is employed. Implementation of DE requires the determination of six fundamental issues: DE step size function, crossover probability, the number of population, initialization, termination and evaluation function. Generally DE step size (F) varies in the interval (0, 2). A good initial guess to F is in the interval (0.5, 1). Crossover probability (CR) constants are generally chosen from the interval (0.5, 1). If the parameter is co-related, then high value of CR work better, the reverse is true for no correlation [8]. In the present study, a population size of NP=20, generation number G=200, step size F=0.8 and crossover probability of CR =0.8 have been used [3, 5]. Optimization is terminated by the prespecified number of generations for DE. Simulations were conducted on a Pentium 4, 3 GHz, 504 MB RAM computer, in the MATLAB 7.0.1 environment. The solver options used in the paper are, Variable step type, ode23s (stiff/Mod. Rosenbroc) solver, with a maximum time step of one cycle of the fundamental frequency. The optimization processes is run 20 times and the best values of PSS parameters obtained by the DE algorithm among the 20 runs is given below:

KP=20.4573, T1=0.2000, T2=0.1500, T3=0.2000, T4=0.0280 5.2

Simulation Results

The controllers are designed at nominal operating conditions for the system subjected to one particular severe disturbance (3-phase fault). To show the robustness of the proposed design approach, different operating conditions and contingencies are considered for the system with and without controller. In all cases, the optimized parameters obtained for the nominal operating condition given above are used as the controller parameters. Also, the simulation results are compared with a conventional power system stabilizer [1]. Three different operating conditions (nominal, light and heavy) are considered and simulation studies are carried out under different fault disturbances and fault clearing sequences. The response without controller is shown with dotted lines with legend ‘NC’; the response with conventional power system stabilizer is shown with dashed line with legend ‘CPSS’ and the response with proposed DE optimized PSS is shown with solid lines with legend ‘DEPSS’. A 3-cycle 3-phase fault is applied at the middle of one transmission line at the nominal operating conditions (Pe = 0.75 pu, δ0 = 45.370). The fault is cleared after 3cycles and the original system is restored. The system response for the above contingency is shown in Figs. 3-5. It is also clear from Figs. that, the proposed DE

64

S.S. Tripathy and S. Panda -3

4

x 10

Δ ω, p.u.

2 0 -2 NC -4

0

1

2

3 Time, sec

CPSS 4

DEPSS 5

6

Fig. 3. Speed deviation response of for 3-cycle 3-phase fault at middle of transmission line with nominal loading condition

55

δ, deg

50 45 40 35

NC 0

1

2

3 Time, sec

CPSS 4

DEPSS 5

6

Fig. 4. Power angle response of for 3-cycle 3-phase fault at middle of transmission line with nominal loading condition

optimized PSS outperform the conventional PSS from dynamic performance point of view. The power system oscillations are quickly damped out with the application of proposed PSS. To test the robustness of the controller to operating condition and fault clearing sequence, the generator loading is changed to light loading condition (Pe = 0.5 pu, δ0 = 22.90), and a 3-cycle, 3-phase fault is applied at Bus2. The fault is cleared by opening both the lines. The lines are reclosed after 3-cycles and original system is restored. The system response is shown in Fig. 6 which shows the robustness of proposed PSS to operating conditions and fault clearing sequence.

Tuning of PSS Employing Differential Evolution Optimization Algorithm

65

1800 1600

P L, MW

1400 1200 1000 800 600 400

NC 0

1

2

CPSS

3 Time, sec

DEPSS

4

5

6

Fig. 5. Tie-line power flow response of for 3-cycle 3-phase fault at middle of transmission line with nominal loading condition -3

6

x 10

Δ ω, p.u.

4 2 0 -2 -4 -6

NC 0

1

2

3 Time, sec

CPSS 4

DEPSS 5

6

Fig. 6. Speed deviation response of for 3-cycle 3-phase fault at bus-2 cleared by both line outage with light loading condition

The effectiveness and robustness of proposed PSS is also verified under heavy loading condition (Pe = 1.0 pu, δ0 = 60.70). A 3-cycle 3-phase fault is applied near the end of one of the transmission line near infinite bus at t=1.0 s. The fault is cleared by opening the faulty line and the line is reclosed after 3-cycles. The system response is shown in Fig. 7. It can be clearly seen from Figs. 13-15 that, for the given operating condition and contingency, the system is unstable without control. Stability of the system is maintained and power system oscillations are effectively damped out with the application of conventional PSS. The proposed PSS provides the best performance and outperform the conventional PSS by minimizing the transient errors and quickly stabilizes the system.

66

S.S. Tripathy and S. Panda

Δ ω, p.u.

0.01

0

-0.01 NC 0

1

CPSS

DEPSS

2

3 Time, sec

4

5

6

Fig. 7. Speed deviation response of for 3-cycle 3-phase fault near infinite bus at end of one transmission line cleared by one line outage with light heavy condition

6 Conclusion In this paper, power system stability enhancement by power system stabilizer is presented. For the proposed controller design problem, a non-liner simulation-based objective function to increase the system damping was developed. Then, differential evolution optimization algorithm is implemented to search for the optimal controller parameters. The effectiveness of the proposed controller, for power system stability improvement, is demonstrated by a weakly connected example power system subjected to different severe disturbances. The dynamic performance of proposed PSS has also been compared with a conventionally designed PSS to show its superiority. The non-linear simulation results presented under wide range of operating conditions; disturbances at different locations as well as for various fault clearing sequences, show the effectiveness and robustness of the proposed DE optimized PSS controller and their ability to provide efficient damping of low frequency oscillations..

References 1. Kundur, P.: Power System Stability and Control. McGraw-Hill (1994) 2. Kundur, P., Klein, M., Rogers, G.J., Zywno, M.S.: Application of power system stabilizers for enhancement of overall system stability. IEEE Trans. Power Syst. 4, 614–626 (1989) 3. Panda, S.: Robust coordinated design of multiple and multi-type damping controller using differential evolution algorithm. Int. J. Elect. Power and Energy Syst. 33, 1018–1030 (2011) 4. Panda, S.: Multi-objective PID Controller Tuning for a FACTS-based damping Stabilizer using Non-dominated Sorting Genetic Algorithm-II. Int. J. Elect. Power and Energy Syst., doi:10.1016/j.ijepes.2011.06.002 5. Panda, S.: Differential evolution algorithm for SSSC-based damping controller design considering time delay. Journal of the Franklin Institute, doi:10.1016/j.jfranklin. 2011.05.011 6. Panda, S.: Multi-objective evolutionary algorithm for SSSC-based controller design. Electric Power System Research 79, 937–944 (2009)

Tuning of PSS Employing Differential Evolution Optimization Algorithm

67

7. Panda, S., Padhy, N.P.: Comparison of particle swarm optimization and genetic algorithm for FACTS-based controller design. Applied Soft Computing 8, 1418–1427 (2008) 8. Stron, R., Price, K.: Differential evolution – a simple and efficient adaptive scheme for global optimization over continuous spaces. J. Global Optim. 11, 341–359 (1997) 9. Gamperle, R., Muller, S.D., Koumoutsakos, P.: A parameter study for differential evolution. In: WSEAS Int. Conf. on Adv. Intell. Syst. Fuzzy Syst. Evolut. Comput., pp. 293–298 (2002) 10. Zaharie, D.: Critical values for the control parameters of differential evolution algorithms. In: Proceedings of the Eighth International Conference on Soft Computing, pp. 62–67 (2002) 11. Vesterstrom, J., Thomsen, R.: A comparative study of differential evolution, particle swarm optimization, and evolutionary algorithms on numerical benchmark problems. In: Proceedings of IEEE Congress on Evolut. Comput., CEC 2004, vol. 2, pp. 1980–1987 (2004) 12. SimPowerSystems 5.2.1, http://www.mathworks.com/products/simpower/

Logistic Map Adaptive Differential Evolution for Optimal Capacitor Placement and Sizing Kamal K. Mandal1,*, Bidishna Bhattacharya2, Bhimsen Tudu1, and Niladri Chakraborty1 1

Jadavpur University, Department of Power Engineering, Kolkata-700098, India [email protected] 2 Techno India, Salt lake, Kolkata, India

Abstract. This paper presents a new adaptive differential evolution technique based on logistic map for optimal distribution placement and sizing. The parameters of differential evolution that need to be selected by the user are the key factors for successful operation DE. Choosing suitable values of parameters are difficult for DE, which is usually a problem-dependent task. Unfortunately, there is no fix rule for selection of parameters. The trial-and-error method adopted generally for tuning the parameters in DE requires multiple optimization runs. Even this method can not guarantee optimal results every time and sometimes it may lead to premature convergence. The proposed method combines differential evolution with chaos theory for self adaptation of DE parameters. The performance of the proposed method is demonstrated on a sample test system. It is seen that the proposed method can avoid premature convergence and provides better convergence characteristics. The results obtained by the proposed methods are compared with other methods. The results show that the proposed technique is capable of producing comparable results. Keywords: Capacitor placement, Loss reduction, Voltage profile, distribution systems, chaos theory.

1 Introduction Capacitor placement plays a very important role in distribution system planning and operation. Capacitors have been very commonly used in distribution systems to provide reactive power compensation. They are used to reduce power losses, to improve power factor and to maintain voltage profile with acceptable limits. The objective in capacitor placement problem is to minimize system losses while satisfying various operating constraints under a certain load pattern. Significant research activity on optimal capacitor placement can be traced back to the 1950’s when the famous “two-thirds” rule was developed. According to this rule, a capacitor rated 2/3 of the total peak reactive demands needs to be installed at a distance of 2/3 along the total feeder length away from the substation for an optimal loss reduction. This rule is still being used as a recommended rule of thumb by many *

Corresponding author.

B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 68–76, 2011. © Springer-Verlag Berlin Heidelberg 2011

Logistic Map Adaptive Differential Evolution

69

utilities [1]. Lee el al [2] objectively criticized the rule and it was shown that it can be misleading in terms of results accuracy. Grainger et al, [3] presented some better methods to solve the problem using shunt as well fixed and switched capacitor assuming non-uniform load profile. The optimal capacitor placement problem is a complex combinatorial optimization problem and several optimization techniques and algorithm have been applied over the years to solve it. These methods include dynamic programming [4], heuristic numerical algorithm [5], genetic algorithm [6] fuzzyreasoning method [7], particle swarm optimization technique [8]. Recently, a new method based on plant growth algorithm was proposed by Wang et al [9]. Differential Evolution (DE) is one of the most recent population-based stochastic evolutionary optimization techniques. Storn and Price first proposed DE in 1995 [10] as a heuristic method for minimizing nonlinear and non-differentiable continuous space functions. Several This paper proposes an adaptive differential evolution (DE) technique using chaos theory to solve the problem of distribution capacitor planning. The feasibility of the proposed method is verified on a sample test system. The results have been compared with other evolutionary methods and it is found that it can produce comparable results.

2 Problem Formulation The objective of capacitor placement problem is to minimize the total annual cost of the system while satisfying some operating constraints under a certain load pattern. The mathematical model of optimal capacitor placement problem can expressed as follows: min F = min (COST )

(1)

where COST includes cost of power loss and capacitor placement. The voltage magnitude at each bus must be maintained within its limits and is expressed as Vmin ≤ Vi ≤ Vmax

(2)

Where Vi is the voltage magnitude of i th bus i, Vmin and Vmax are the minimum and maximum bus voltage limits respectively.

Fig. 1. Single -line diagram of a main feeder

70

K.K. Mandal et al.

A set of simplified feeder-line flow formulation is assumed for simplicity. Considering the one-line diagram shown depicted in Fig.1, the following set of equations may be used for power flow calculation [Sundharajan and Pahwa 1994]. Pi +1 = Pi − PLi +1 − Ri ,i +1 ×

(P

2 2 i + Qi 2 Vi

)

(

P 2 + Qi2 Qi +1 = Qi − QLi +1 − X i,i +1 × i 2 Vi Vi +1

2

= Vi

2

(

(3)

)

(

)

− 2 Ri,i +1.Pi + X i,i +1.Qi + Ri2,i +1 + X i2,i +1

(4)

) (P

2 2 i + Qi 2 Vi

)

(5)

where Pi and Qi are the real and reactive powers flowing out of i th bus respectively. PLi , Q Li are the real and reactive load powers at the i th bus respectively. The resistance and reactance of the line section between buses i and i+1 are denoted by Ri ,i +1 and X i ,i +1 respectively. The power loss of the line section connecting buses i and i + 1 can be calculated as

(

P 2 + Qi2 PLoss (i, i + 1) = Ri ,i +1 × i 2 Vi

)

(6)

The total power loss of the feeder PT , Loss may then be determined by summing up the losses of all line sections of the feeder. The loss is given by

PT , Loss =

n −1

P

Loss

(i, i + 1)

(7)

i =0

The principle of placing compensating capacitor along distribution feeders is to lower the total power loss and transport the bus voltages within their specified limits while minimizing the total cost. Considering the practical capacitors, there exists a finite number of standard sizes which are integer multiples of the smallest size Q0c . Besides, the cost per kVAr varies from one size to another. In general, capacitors of larger size have lower unit prices. The available capacitor size is usually limited to Q cmax = LQ 0c

(8)

where L is an integer. Therefore, for each installation location, there are L capacitor sizes Q0c ,2Q0c ,..., LQ0c available. Let K1c , K 2c ,..., K Lc be their corresponding equivalent annual cost per kVAr. Therefore, the total annual cost function due to capacitor placement and power loss may be found as

{

}

{

}

n

COST = K P PT , Loss +

K Q c i

i =1

c i

(9)

Logistic Map Adaptive Differential Evolution

71

where K P is the equivalent annual cost per unit of power loss in $/ (kW – year) and, i = 1, 2, . . . , n are the indices of buses selected for compensation. The bus reactive compensation power is limited to n

Qic ≤

Q

(10)

Li

i =1

3 Overview of Differential Evolution DE or Differential Evolution belongs to the class of evolutionary algorithms [10] that include Evolution Strategies (ES) and conventional genetic algorithms (GA). DE offers several strategies for optimization [11]. The version used here is the DE/rand/1/bin, which is described by the following steps. 3.1 Initialization

The optimization process in DE is carried with four basic operations: initialization, mutation, crossover and selection. The algorithm starts by creating a population vector P of size NP composed of individuals that evolve over G generations. Each individual Xi is a vector that contains as many elements as the problem decision variable. The population size NP is an algorithm control parameter selected by the user. Thus,

[

P (G ) = X i(G ) ,.........., X N(G ) P

(G )  X i(G ) =  X1(,Gi ) ,............. X D , i   i = 1,..............N P

]

(11)

T

(12)

The initial population is chosen randomly in order to cover the entire searching region uniformly. A uniform probability distribution for all random variables is assumed in the following as X (j0, i) = X min +σ j

j

(X

max j

− X min j

)

(13)

where i = 1,....................N P and j = 1,....................D ; and X max are the Here D is the number of decision or control variables, X min j j

lower and upper limits of the j the decision variables and σ j ∈ [ 0 , 1 ] is a uniformly distributed random number generated anew for each value of j. X (j0,i) is the j th parameter of the i th individual of the initial population. 3.2

Mutation Operation

Several strategies of mutation have been introduced in the literature of DE. The essential ingredient in the mutation operation is the vector difference. The mutation operator

72

K.K. Mandal et al.

creates mutant vectors ( Vi ) by perturbing a randomly selected vector ( X k ) with the difference of two other randomly selected vectors ( X l and X m ) according to: (G )  Vi(G ) = X (Gk) + f M  X l(G ) − X m   

(14)

where X k , X l and X m are randomly chosen vectors ∈ [ 1 ,......................, N P ] and k ≠ l ≠ m ≠ i. The mutation factor f M that lies within [0, 2] is a user chosen parameter used to control the perturbation size in the mutation operator and to avoid search stagnation. 3.3 Crossover Operation

In order to extend further diversity in the searching process, crossover operation is performed. The crossover operation generates trial vectors ( U i ) by mixing the parameter of the mutant vectors with the target vectors. For each mutant vector, an index q ∈ [ 1,.................N P ] is chosen randomly using a uniform distribution and trial vectors are generated according to: V (G )  j, i (G )  U j, i =   (G )  X j, i 

, if η j ≤ C R or j = q

(15)

, otherwise

where i = 1,.................., N P and j = 1,................., D ; η j is a uniformly distributed random number within [0, 1] generated anew for each value of j. The crossover factor C R ∈ [ 0, 1] is a user chosen parameter that controls the diversity of the population. X (jG, i ) , V j(,Gi ) and U (jG, i ) are the j th parameter of the i th target vector, mutant vector

and trial vector at generation G respectively. 3.4 Selection Operation

Selection is the operation through which better offspring are generated. The evaluation (fitness) function of an offspring is compared to that of its parent. Thus, if f denotes the cost (fitness) function under optimization (minimization), then X i(G +1)

( ) (

U (G ) , if f U (G ) ≤ f X (G ) i i  i =  (G )  X i , otherwise

)

(16)

The optimization process is repeated for several generations. The iterative process of mutation, crossover and selection on the population will continue until a userspecified stopping criterion, normally, the maximum number of generations allowed, is met.

Logistic Map Adaptive Differential Evolution

73

4 Hybrid Differential Evolution Using Chaos Theory Optimization algorithms based on chaos theory are stochastic search methodologies and are different from the existing evolutionary algorithms. Evolutionary algorithms use the concepts of bio-inspired genetics and natural evolution. On the other hand, optimization techniques using chaos theory based on ergodicity, stochastic properties, and irregularity [12]. Chaotic sequences display an unpredictable long-term behavior due to their sensitiveness to initial conditions. This feature can be utilized to track the chaotic variable as it travels ergodically over the searching space. Crossover ratio and mutation factors are the two important user defined parameters and successful operation of differential evolution is heavily dependent on these two parameters. This paper utilizes chaotic sequence for automatic adjustment of DE parameters. This helps to escape from local minima and improves global convergence. One of the simplest dynamic systems evidencing chaotic behavior is the iterator called the logistic map [13] and can be described by the following equation. y (t ) = μ ⋅ y (t − 1) ⋅ [1 − y (t − 1)]

(17)

Where t is the sample and µ is control parameter, 0 ≤ μ ≤ 4 . The behavior of the system described by (17) is greatly changed with the variation of µ. The value of µ determines whether y stabilizes at a constant size, oscillates between a limited sequence of sizes, or behaves chaotically in an unpredictable pattern. Equation (17) is deterministic displaying chaotic dynamics when μ = 4 and y (0 )∉{0, 0.25, 0.5, 0.75, 1} . In this case, y(t) is distributed in the range of (0,1) provided the initial . y (0 )∈ (0,1) . The values of the parameters mutation factor (fM) and cross over ratio (CR) can be modified using (16) as follows: f m (G ) = μ ⋅ f m (G − 1) ⋅ [1 − f m (G − 1)]

(18)

C R (G ) = μ ⋅ C R (G − 1) ⋅ [1 − C R (G − 1)]

(19)

where G is the current iteration number.

5 Structure of Solutions In this section, an algorithm based on a novel hybrid DE for optimal solution of capacitor placement problem is described. For any population based algorithm like PSO, the representation of individuals and their elements is very important. For the present problem, it is the candidate bus where the capacitor is to be placed. The algorithm

[

starts with the initialization process. Let P ( 0) = X 1( 0) , X 2(0) ,  X k( 0) ,  , X N(0)

be the initial population of N p number of particles.

P

]

For a system of n number of

candidate buses, position of k th individual is of n-dimension and can be represented by

74

K.K. Mandal et al.

[

c (0 ) X k(0 ) = Qkc1(0 ) , Qkc2(0 ) ,  Qkjc (0 ) ,  , Qkn

]

(20)

The element Qkjc (0 ) represents a randomly selected capacitance from the available size satisfying the constraints given by (8).

6 Simulation Results The proposed algorithm was implemented using in house Matlab code on 3.0 MHz, 2.0 GB RAM PC. To demonstrate the effectiveness and feasibility of the proposed algorithm, it was applied on a sample test system. A population size of 30 and maximum iteration number of 300 was chosen after 50 trial runs for optimal solutions. The test system [7] under consideration consists of a 23 kV, 9 section feeder. Feeder impedance data, three phase load, available capacitor size with their corresponding annual cost and other data are taken from [7] and not shown here due to page limitation. The equivalent unit cost per unit of power loss considered for the present problem is $168/(kW-year). The limits on bus voltages are as follows: Vmin = 0.90 p.u. Vmax = 1.10 p.u.

It is considered that all the buses were available for compensation. The annual costs, system power loss both before and after compensation, capacitor addition at the desired location are shown in Table 1. It is seen from Table 1 that voltage profile for all the buses are within the system limits. The annual cost is $114,757 while the system power loss is 0.6731 MW in comparison with uncompensated cases where the annual cost is $131,675 and power loss is 0.7837 MW. The convergence characteristic for cost is shown in Fig.2. Table 1. Results including Voltage profile, Annual cost, Capacitor and Power Loss Bus No. 0 1 2 3 4 5 6 7 8 9 Total cap. size (Mvar) Total Loss (MW) Annual cost in ($/year) CPU time (sec)

Uncompensated Voltage (p.u) 1 0.9929 0.9874 0.9634 0.9619 0.9480 0.9072 0.8890 0.8587 0.8375

Placed (Qc) (kVar) 0 0 4050 1200 1650 0 1200 0 450 450

0.7837 131,675 80.23

Compensated Voltage (p.u) 1.0000 1.0000 1.0050 0.9934 0.9832 0.9614 0.9554 0.9406 0.9177 0.9011 9.00 0.6731 114,757

Logistic Map Adaptive Differential Evolution

75

. Fig. 2. Convergence Characteristics for Cost

The result is also compared with other methods like fuzzy reasoning [7], DE, GA, ACSA [14] and is shown in Table 2. It is seen from the Table 6, that the proposed method can avoid premature convergence successfully and produce better results. Table 2. Comparison of results with different methods

Total Loss (MW) Annual cost in ($/year)

Fuzzy Reasoning 0.7048

DE

GA

ACSA

0.6763

0.6766

0.6753

Proposed Method 0.6734

119,420

115,471

115,572

115,395

114,757

7 Conclusions Capacitor placement is one of the important issues in modern day power system operation. The basic objective is to reduce power losses as well as to improve voltage profile. In this paper, an algorithm based on a novel hybrid differential evolution technique has been successfully applied for solving optimal capacitor placement problem to avoid premature convergence. To evaluate the performance of the proposed algorithm, it has been applied on a sample test system. The results obtained by the proposed method have been compared with other population based algorithms like fuzzy reasoning, DE, GA, ACSA. The results show that the proposed algorithm is indeed capable of obtaining good quality solution efficiently in case of capacitor placement problems.

76

K.K. Mandal et al.

References [1] Ng, H.N., Salama, M.M.A., Chikhani, A.Y.: Classification of capacitor allocation techniques. IEEE Transactions on Power Delivery 15(1), 387–392 (2000) [2] Lee, S.H., Grainger, J.J.: Optimum placement of fixed and switched capacitors on primary distribution feeder. IEEE Trans. on Power Apparatus and Systems 100(1), 345–352 (1981) [3] Grainger, J.J., Lee, S.H.: Capacity release by shunt capacitor placement on distribution feeder: A new voltage-dependent model. IEEE Trans. on Power Apparatus and Systems 101(5), 1236–1244 (1982) [4] Duran, H.: Optimum number, location, and size of shunt capacitors in radial distribution feeder: A dynamic programming approach. IEEE Trans. on Power Apparatus and Systems 87(9), 1769–1774 (1983) [5] Baghzouz, Y., Ertem, S.: Shunt capacitor sizing for radial distribution feeders with distorted substation voltages. IEEE Trans. on Power Delivery 5, 650–657 (1990) [6] Sundharajan, S., Pahwa, A.: Optimal selection of capacitors for radial distribution systems using genetic algorithm. IEEE Trans. Power Systems 9(3), 1499–1507 (1994) [7] Su, C.T., Tasi, C.C.: A new fuzzy reasoning approach to optimum capacitor allocation for primary distribution systems. In: Proc. 1996 IEEE on Industrial Technology Conference, pp. 237–241 (1996) [8] Yu, X., Xiong, X., Wu, Y.: A PSO based approach to optimal capacitor placement with harmonic distortion consideration. Electric Power System Research 71, 27–33 (2004) [9] Wang, C., Cheng, H.Z.: Reactive power optimization by plant growth simulation algorithm. IEEE Trans. on Power Systems 23(1), 119–126 (2008) [10] Storn, R., Price, K.: Differential Evolution - A simple and efficient adaptive scheme for global optimization over continuous spaces. Journal of Global Optimization 11, 341–359 (1997) [11] Mallipeddi, R., Suganthan, P.N., Pan, Q.K., Tasgetiren, M.F.: Differential evolution algorithm with ensemble of parameters and mutation strategies. Applied Soft Computing 11(2), 1679–1696 (2011) [12] Coelho, L.S., Mariani, V.C.: Combining of Chaotic Differential Evolution and Quadratic Programming for Economic Dispatch Optimization with Valve-Point Effect. IEEE Transaction on Power Systems 21(2), 989–996 (2006) [13] May, R.: Simple mathematical models with very complicated dynamics. Nature 261, 459– 467 (1976) [14] Su, C.T., Chan, C.F., Chiou, J.P.: Capacitor placement in distribution system employing ant colony search algorithm. Electric Components and Systems 33, 931–946 (2005)

Application of an Improved Generalized Differential Evolution Algorithm to Multi-objective Optimization Problems Subramanian Ramesh1, Subramanian Kannan2, and Subramanian Baskar3 1

Arulmigu Kalasalingam College of Engineering, Krishnankoil - 626 126, India 2 Kalasalingam University, Anand Nagar, Krishnankoil-626 126, India 3 Thiagarajar College of Engineering, Madurai - 625 015, India [email protected], [email protected], [email protected]

Abstract. An Improved Multiobjective Generalized Differential Evolution ( IGDE3) approach is proposed in this paper. For maintaining good diversity, the concepts of Simulated Binary Crossover (SBX) based recombination and Dynamic Crowding Distance (DCD) are implemented in GDE3 algorithm. The proposed approach is a p p l i e d to different sets of classical test problems suggested in the MOEA literature to validate the performance of the I-GDE3. Later, t h e proposed approach i s i m p l e m e n t e d t o R e a c t i v e P o w e r P l a n n i n g (RPP) problem. The objective functions are minimization of combined operating and VAR allocation cost and bus voltage profile improvement. The performance of the proposed approach is tested in standard IEEE 30-bus test systems. The performance of I-GDE3 is compared with respect to multi- objective performance measures namely gamma, spread, minimum spacing and Inverted Generational Distance (IGD). The results show the effectiveness of I-GDE3 and confirm its potential to solve the multi-objective problems.

1 Introduction Multi-objective optimization problems, unlike a single objective optimization problem, do not necessarily have an optimal solution that minimizes all the multiobjective functions simultaneously. Often, different objectives may conflict each other and the optimal parameters of some objectives usually do not lead to optimality of other objectives (sometimes make them worse). Evolutionary Algorithms (EAs) can find multiple optimal solutions in single simulation run due to their population approach. The main advantage of evolutionary algorithms (EAs) in solving multi-objective optimization problems is their ability to find multiple Pareto-optimal solutions in one single run. Recently, a number of multiobjective evolutionary algorithms (MOEAs) have been suggested by [5, 25, 26, 28]. Most of these algorithms were developed taking into consideration of two common goals, namely fast convergence to the Pareto-optimal front and good distribution of solutions along the front. Each algorithm employs a unique combination of specific techniques to achieve these goals. MOEA/D [26] decomposes a multiobjective problem into a number of scalar optimization subproblems and B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 77–84, 2011. © Springer-Verlag Berlin Heidelberg 2011

78

S. Ramesh, S. Kannan, and S. Baskar

optimizes them simultaneously. PAES proposed by [13], uses a histogram-like density measure over a hyper-grid division of the objective space. Deb [5] proposed NSGA-II approach which incorporates elitist and crowding approaches. The differential evolution (DE) algorithm has been found to be successful in single objective optimization problems [21]. Recently there are several attempts to extend the DE to solve multi-objective problems. One approach is presented to optimize train movement by tuning fuzzy membership functions [4]. A Pareto-frontier Differential Evolution algorithm (PDE) to solve multi-objective problem by incorporating Pareto dominance is proposed by [1]. This PDE is also extended [2] with self-adaptive crossover and mutation. An extended DE approach for solving multi-objective optimization problems by incorporating a non-dominated sorting and ranking selection scheme of NSGA-II is presented in [17]. In the literature, the third evolution step of Generalized Differential Evolution (GDE3) has been reported in [14] as an extension of DE for constrained multi-objective optimization. In GDE3, reproduction part of the algorithm is similar to DE [21] and selection and diversity are similar to NSGA-II [5]. Another version of MODE algorithm is found and is applied to industrial styrene reactor [7]. As an extension of this MODE, the concept of elitism is incorporated with the MODE algorithm [6] and it is named as Elitist MODE (EMODE) which is applied to Purified Terephthalic Acid (PTA) Oxidation Process. A hybrid version of MODE (H-MODE) which consists of MODE with sequential simplex based local search technique is employed successfully for optimizing the Adiabatic Styrene Reactor and Polyethylene Terephthalate (PET) reactor [8,9]. Wang et al. proposes a self-adaptive differential evolution algorithm which incorporates the concept of Pareto dominance to solve multi-objective optimization problem [23]. Recently, the NSGA-II algorithm is used in various power system problems such as economic dispatch [12] and Generation Expansion Planning [11]. Some successful application of EAs to optimal Reactive Power Planning (RPP) has been reported in the literature [15, 3, 27], where minimizing voltage differences have been considered as a multi-objective in addition to cost minimization. The objective of this paper is to solve the multi-objective RPP problem using an Improved Multiobjective Generalized Differential Evolution (I-GDE3) algorithm. This algorithm replaces the crowding distance operator [15] in the original GDE3 algorithm by means of a diversity maintenance strategy which is based on Dynamic Crowding Distance (DCD) proposed by [17]. To validate the performance of the algorithm, performance metrics such as gamma, delta, minimum spacing and inverted generational distance are used.

2 Multi-objective Differential Evolution Storn and Price proposed an evolutionary algorithm called Differential Evolution (DE) to solve real-parameter optimization problems [21]. DE uses a simple mutation operator based on differences between pairs of solutions (called vectors) with the aim of finding a search direction based on the distribution of solutions in the current population. DE also utilizes a steady-state-like replacement mechanism, where the newly generated offspring (called trial vector) competes only against its corresponding parent (old object vector) and replaces it if the offspring has a higher fitness value.

Application of an Improved Generalized Differential Evolution Algorithm

79

In order to apply the DE strategy for solving multi-objective optimization problems, the original scheme has to be modified since the solution set of a problem with multiple objectives does not consist of a single solution (as in global optimization). Instead, in multi-objective optimization, the intention is to find a set of different solutions (the so-called Pareto optimal set). Two design aspects should be considered for DE algorithm for extending to multi-objective optimization. The first is to promote diversity among the individuals in the population which can be achieved by means of crowding distance measure and the second is to select the individuals in the population through non-dominated sorting approach proposed by Deb [5]. In the subsequent sections, proposed MODE algorithm is discussed. 2.1 Implementation of I-GDE3 Algorithm There are different types of MODE algorithm available in the literature. They differ mainly on the type of selection mechanism, mutation strategy and diversity maintenance. In the selection mechanism, MODE can be further classified based on Pareto dominance [1,2,4] and Pareto ranking concepts [6, 7, 8, 9, 14, 17]. In mutation strategy, MODE vary on the type of the criterion to select one of the individuals to be used in the mutation operator (called donor vector), the number of differences computed also in the mutation operator and, finally, in the recombination operator chosen. Several types of mutation strategies were used in MODE namely DE/current to rand/1/bin [1, 2, 17], DE/rand/1/bin [6, 7, 8, 9, 14] and DE/best/1/bin [24]. In diversity maintenance, two major approaches were used. One is based on fitness sharing [4] and the other is based on crowding distance [9, 14]. This paper proposes Simulated Binary Crossover (SBX) as a recombination operator [5] in the MODE algorithm [19]. Besides typical parameters used in EAs (number of individuals and number of iterations), two parameters adopted in I-GDE3 are: crossover index (ηc) and step size (F). Theηc controls the influence of the parents in the generation of the offspring. Higher value of “ηc” means higher probability for creating solutions nearer to the parents and a smaller value of “ηc” allows creating solutions away from the parents. “F” scales the influence of the set of pairs of solutions selected to calculate the mutation value. It is of great importance for Pareto front with good convergence and diversity characteristics with respect to true Pareto front. A good diversity can give decisionmakers more reasonable and efficient selections. The diversity maintenance strategy (DMS) is often realized in the process of population maintenance, which uses a truncation operator to wipe off individuals when the number of non-dominated solutions exceeds population size. In this paper, a DMS which is based on dynamic crowding distance (DCD) is used [16]. 2.2 I-GDE3 Algorithm The following steps can be adopted for the implementation of proposed MODE algorithm. Step1: Identify the control variables for the problem Step2: Select the parameters like population size, maximum number of iterations, crossover index (ηc) and mutation parameter (F).

80

S. Ramesh, S. Kannan, and S. Baskar

Step3: Generate initial parent population (P) Step4: Evaluate objective functions for initial population Step5: Set the generation count Step6: a. Perform mutation in the parents to generate mutated parents (Qm) b. Perform recombination using SBX crossover (of P size), for the entire mutated parents created from step 6a (QC). Step7: Perform non-dominated sorting. (i.e., sorting the population according to each objective function value in ascending order of magnitude) for the combined parent and offspring population ( P ∪ QC ). Step8: Calculate DCD for the combined population based on the algorithm discussed in [16]. Step9: Increment the iteration count and repeat the steps from 6 to 9 until the count reaches the maximum number of iterations.

3 Simulation Results The I-GDE3 algorithm was implemented using MATLAB version 7.4, on an IBM PC with Pentium dual core processor having 1.86GHz clock speed and 1 GB of RAM. 3.1 Standard Test Problems Different sets of classical test problems suggested in the MOEA literature are used to validate the performance of the I-GDE3. The results are sensitive to algorithm parameters. Hence, it is required to perform repeated simulations to find suitable values for the parameters. Optimal parameter combinations for different methods are experimentally determined by conducting series of experiments with different parameter settings before conducting actual runs to obtain the results. The crossover index (ηc) is selected between 1 and 10, in steps of 1 and for each ηc performance is analyzed. It is found that ηc = 3, produces the best results. During simulation, ‘F’ is varied in the range 0.1 to 1 in steps of 0.1. It is identified with a step length of 1 is more suitable for better convergence characteristics. Table 1. Mean and Variance Values of the Convergence (γ) and Divergence (Δ) Metrics for constrained optimization problems & KUR problem Problem OSY SRN TNK KUR

Algorithms NSGA II I-GDE3 NSGA II I-GDE3 NSGA II I-GDE3 NSGA II I-GDE3

Convergence Mean Variance 1.5615 0.2232 0.1249 1.1782 0.2397 0.0346 0.0076 0.0588 0.0023 3.59E-4 2.21E-4 0.00145 0.028964 0.000018 0.00018741 0.0021233

Divergence Mean Variance 0.0882 0.7482 0.7709 0.0752 0.3906 0.0452 0.0162 0.1502 0.8481 0.1005 0.0421 0.3640 0.000992 0.4115 0.4936 0.00513

Application of an Improved Generalized Differential Evolution Algorithm

81

The performance of the proposed I-GDE3 algorithm is tested in a benchmark test unconstrained (KUR) optimization and constrained optimization problems such as OSY, SRN and TNK [5]. The performance of I-GDE3 is compared with respect to the multi-objective performance measures (namely convergence and divergence metrics) and it is reported in Table 1. The results show the effectiveness of I-GDE3 and confirm its potential to solve the multi-objective RPP problem. All results have been averaged over 30 independent runs. A result with boldface indicates better value obtained. From the Table 1, it is observed that the performance of I-GDE3 is better than NSGA-II in the standard benchmark test problems. 3.2 Implementation of I-GDE3 to RPP Problem This paper considers a short-term RPP, where the investment is to be performed only once. The first objective function comprises two terms [10, 15]. The first term represents the total cost of energy loss and the second term represents the cost of VAR source installations which has two components, namely, fixed installation cost and cost of the VAR source. The second objective is to minimize the deviations in voltage magnitudes at load bus [3, 27]. In the control variables, the generator bus voltages i.e. Automatic Voltage Regulator operating values are taken as continuous variable, whereas the transformer tap settings and shunt susceptance values are taken as discrete values. The IEEE 30-bus test system is considered as a test system the detailed data are given in [22]. The system has 6 generators, 4 transformers and 9 shunt compensators; therefore, the numbers of variables to be optimized are 19. The bus numbers 1, 2, 5, 8, 11, and 13 are generator bus. The lower and upper limits for voltage magnitude of these buses are 0.95 p.u. and 1.1 p.u., and for the remaining bus lower and upper limits for voltage magnitude of bus are 0.95 p.u. and 1.05 p.u. respectively. The transformer tapping are varied between 0.9 and 1.1 p.u. with the step size of 0.0001 and the shunt capacitors have the rating between 0 and 5 MVAR with each capacitor with the step size of 1 MVAR. The RPP problem is treated as true multi-objective optimization problem where combined operating and investment costs and voltage deviations are optimized simultaneously with the I-GDE3 algorithm. The NSGA-II and I-GDE3 algorithm are applied to solve the RPP problem and the best pareto-front obtained among 15 simulation runs is represented in fig. 1. In some problems, the true pareto-front may not be available. For that case, the determination of reference pareto-front is needed. In this paper, Covariance Matrix Adapted Evolution Strategy (CMA-ES) technique [18] with conventional weighted sum approach is used as reference pareto-front since for a RPP problem there is no true pareto-front reported in the literature. So the performance metrics have been calculated for NSGA-II, MNSGA-II [20] and I-GDE3 (by comparing with the reference pareto-front generated by CMA-ES).The statistical results of performance metrics such as mean and standard deviation (σ) in 15 simulation runs are represented in Table 2 and the bold numbers indicate the best mean values. From Table 2, it is observed that except for minimum spacing, the performance of I-GDE3 is better than NSGA-II and MNSGA-II with respect to the mean values of metrics.

82

S. Ramesh, S. Kannan, and S. Baskar Table 2. Statistical Comparison of Performance Metrics IEEE 30-bus Performance Measures Mean σ Mean σ Mean σ Mean σ

Gamma (γ) Spread (Δ) Minimum Spacing (Sm) IGD

NSGA-II

I-GDE3

0.1146 0.0854 0.0675 0.0711 0.0376 0.0590 0.2311 0.0362

MNSGA-II

0.1118 0.0765 0.0621 0.0638 0.0412 0.0503

0.1118 0.1071 0.0629 0.0668 0.0542 0.0503 0.2291 0.0563

0.2281 0.0473

Bus Voltage Deviation in p.u.

1.5 I-GDE3 NSGA-II MNSGA-II Reference Pareto-front 1

0.5

0

1

2

3

4 Cost in $/year

5

6

7 x 10

6

Fig. 1. Comparison of Pareto-Optimal Set for IEEE 30-bus

4

Conclusion

In this paper, Improved Generalized Differential Evolution (I-GDE3) is applied to standard multi-objective test problem and reactive power planning problem (RPP). In RPP, conflicting objectives such as combined operating and VAR allocation cost minimization and bus voltage profile improvement are considered. The IEEE 30-bus system is considered as a test system. By using conventional weighted sum method with Covariance Matrix Adapted Evolution Strategy (CMA-ES), a reference paretofront is generated. The statistical multi-objective performance measures such as gamma, spread, minimum spacing and Inverted Generational Distance are considered for validating the improved performance of I-GDE3. From the simulation results, it was observed that I-GDE3 performs better in most of the performance measures when compared to NSGA-II for both standard test problem and RPP problem. The simulation result clearly shows that I-GDE3 algorithm is certainly more suitable for solving multi-objective RPP problems.

Application of an Improved Generalized Differential Evolution Algorithm

83

Acknowledgments. The authors gratefully acknowledge the management of Kalasalingam University, Krishnankoil, Thiagarajar College of Engineering, Madurai, Tamilnadu, India. The authors are also thankful to the Department of Science and Technology, New Delhi for its support through the project SR/S4/MS/350/06 to the second author.

References 1. Abbass, H.A., Sarker, R., Newton, C.: PDE: A Pareto-frontier Differential Evolution Approach for Multi-objective Optimization Problems. In: Proc. of IEEE Cong. on Evol. Comp., pp. 971–978 (2001) 2. Abbass, H.A.: The Self-Adaptive Pareto Differential Evolution Algorithm. In: Proc.of IEEE Cong. on Evol. Comp., vol. 1, pp. 831–836 (2002) 3. Abido, M.A., Bakhashwain J.M.: A Novel Multi-objective Evolutionary Algorithm for Reactive Power Dispatch Problem. In: ICECS 2003, pp. 1054–1057 (2003) 4. Chang, C.S., Xu, D.Y., Quek, H.B.: Pareto-optimal set based multi-objective tuning of fuzzy automatic train operation for mass transit system. IEE Proceedings on Electric Power Applications 146(5), 577–583 (1999) 5. Deb, K.: Multi-Objective Optimization using Evolutionary Algorithms, 1st edn. John Wiley & Sons, Ltd., Singapore (2001) 6. Gujarathi, A.M., Babu, B.V.: Improved Multi-Objective Differential Evolution (MODE) Approach for Purified Terephthalic Acid (PTA) Oxidation Process. Materials and Manufacturing Processes 24(3), 303–319 (2009) 7. Gujarathi, A.M., Babu, B.V.: Multi-objective Optimization of Industrial Styrene Reactor: Adiabatic and Pseudo-isothermal Operation. Chem. Engg. Science 65(6), 2009–2026 (2010) 8. Gujarathi, A.M., Babu, B.V.: Optimization of Adiabatic Styrene Reactor: A Hybrid MultiObjective Differential Evolution (H–MODE) Approach. Industrial & Engineering Chemistry Research 48(24), 11115–11132 (2009) 9. Gujarathi, A.M., Babu, B.V.: Hybrid Multi-objective Differential Evolution (HMODE) for optimization of Polyethylene Terephthalate (PET) reactor. Int. J. of Bio. Insp. Comp. 2(3/4), 213–221 (2010) 10. Hsaio, Y.T., Chaing, H.D., Liu, C.C., Chen, Y.L.: A Computer Package for Optimal Multiobjective VAR Planning in Large Scale Power Systems. IEEE Trans. on Power Syst. 9(2), 668–676 (1994) 11. Kannan, S., Baskar, S., McCalley, J.D., Murugan, P.: Application of NSGA-II Algorithm to Generation Expansion Planning. IEEE Trans. on Power Syst. 24(1), 454–461 (2009) 12. King, R.T.F.A., Rughooputh, H.C.S.: Elitist multiobjective evolutionary algorithm for environmental/economic dispatch. In: The 2003 Congress on Evolutionary Computation, vol. 2, pp. 1108–1114 (2003) 13. Knowles, J., Corne, D.: The Pareto archived evolution strategy: A new baseline algorithm for multi-objective optimization. In: Proceedings of the 1999 Congress on Evolutionary Computation, pp. 98–105 (1999) 14. Kukkonen, S., Lampinen, J.: GDE3: The third evolution step of Generalized Differential Evolution. In: Proceedings of the 2005 Congress on Evolutionary Computation (CEC 2005), pp. 443–450 (2005) 15. Lai, L.L., Ma, J.T.: Application of Evolutionary Programming to Reactive Power Planning – Comparison with Nonlinear Programming Approach. IEEE Trans. on Power Syst. 12(1), 198–206 (1997)

84

S. Ramesh, S. Kannan, and S. Baskar

16. Luo, B., Zheng, J., Xie, J., Wu, J.: Dynamic Crowding Distance - A New Diversity Maintenance Strategy for MOEAs. In: ICNC 2008, 4th International Conference Natural Computation, vol. 1, pp. 580–585 (2008) 17. Madavan, N.K.: Multiobjective Optimization Using a Pareto Differential Evolution Approach. In: Proceedings of Congress on Evolutionary Computation (CEC 2002), vol. 2, pp. 1145–1150 (2002) 18. Manoharan, P.S., Kannan, P.S., Baskar, S., Willjuice Iruthayarajan, M., Dhananjeyan, V.: Covariance matrix adapted evolution strategy algorithm-based solution to dynamic economic dispatch problems. Engg. Optimization 41(7), 635–657 (2009) 19. Ramesh, S., Kannan, S., Baskar, S.: An improved generalized differential evolution algorithm for multi-objective reactive power dispatch. Engineering Optimization (2011), doi:10.1080/0305215X.2011.576761 20. Ramesh, S., Kannan, S., Baskar, S.: Multi-objective Evolutionary Algorithm based Reactive Power Dispatch with Thyristor Controlled Series Compensators. In: IEEE PEDES 2010 and Power India, pp. 1–5 (2010) 21. Storn, R., Price, K.: Differential evolution-A simple and Efficient Heuristic for Global Optimization over Continuous Spaces. J. Glob. Optim. 11, 341–359 (1997) 22. University of Washington, Power Systems Test Case Archive, http://www.ee.washington.edu/research/pstca/ 23. Wang, Y., Wu, L., Yuan, X.: Multi-objective self-adaptive differential evolution with elitist archive and crowding entropy-based diversity measure. Soft Computing 14(3), 193– 209 (2010) 24. Xue, F., Sanderson, A.C., Graves, R.J.: Modeling and convergence analysis of a continuous multi-objective differential evolution algorithm. In: 2005 IEEE Congress on Evolutionary Computation (CEC 2005), vol. 1, pp. 228–235 (2005) 25. Zhao, S.Z., Suganthan, P.N.: Two-lbests Based Multi-objective Particle Swarm Optimizer. Engineering Optimization 43(1), 1–17 (2011) 26. Zhao, S.Z., Suganthan, P.N., Zhang, Q.: Decomposition Based Multiobjective Evolutionary Algorithm with an Ensemble of Neighborhood Sizes. IEEE Trans. on Evolutionary Computation (accepted) 27. Zhihuan, L., Yinhong, L., Xianzhong, D.: Non-dominated sorting genetic algorithm-II for robust multi-objective optimal reactive power dispatch. IET Gen., Trans. and Dist. 4(9), 1000–1008 (2010) 28. Zhou, A., Qu, B.Y., Li, H., Zhao, S.Z., Suganthan, P.N., Zhang, Q.: Multiobjective evolutionary algorithms: A survey of the state of the art. Swarm and Evolutionary Computation 1, 32–49 (2011)

Enhanced Discrete Differential Evolution to Determine Optimal Coordination of Directional Overcurrent Relays in a Power System Joymala Moirangthem1, Subranshu Sekhar Dash1, K.R. Krishnanand2, and Bijaya Ketan Panigrahi3 1 2

Department of Electrical and Electronics Engineering, SRM University, Tamil Nadu, India Multi-Disciplinary Research Cell, Siksha ‘O’ Anusandhan University, Bhubaneswar, India 3 Department of Electrical Engineering, Indian Institute of Technology, Delhi, India [email protected], [email protected], [email protected], [email protected]

Abstract. This paper presents an enhanced differential evolution technique to solve the optimal coordination of directional overcurrent relays in a power system. The most vital task when installing directional relays on the system is selecting suitable current and time settings such that their fundamental protective functions are met under the requirements of sensitivity, selectivity, reliability and speed. Coordination of directional over current relays can be formulated as a linear or non-linear problem taking into account the discrete value for the time dial settings and the pickup current settings. The results are compared with different algorithms on a test system and are presented in this paper. Keywords: Directional overcurrent relays (DOCRs), Enhanced Discrete Differential Evolution Algorithm (EDDEA), Relay coordination, Pickup current settings (Ip), Time dial settings (TDS).

1 Introduction The basic role of a transmission protection system is to sense faults on lines or at a substation and to rapidly isolate these faults by opening all incoming current paths. It should be very selective in disconnecting the network from service to avoid unnecessary removal of network and also it should reliable. This need has made the protection of system to come up with two types of protection i.e. primary protection and back up protection. The primary relays are designed for speed and minimum disruption of network while back up relay operates more slowly than the primary and affects larger network [1]. The problem of coordination involves computation of time dial setting (TDS) and pick up current (Ip) setting or plug setting. Various techniques have been approached to solve the coordination problem since the manual computation is very tedious and time consuming task. Many researchers report on computing the optimal coordination of directional overcurrent relays by using linear programming techniques such simplex method [2]-[4], dual simplex methods [5]. A detail report has been performed by Birla et al in solving coordination problem by three techniques namely curve-fitting technique, graph theoretical technique and B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 85–93, 2011. © Springer-Verlag Berlin Heidelberg 2011

86

J. Moirangthem et al.

optimization technique [6]. In [7] random search technique to achieve proper coordination with acceptable speed of the primary protection. Stochastic search technique like GA and PSO started applying by many researchers to solve the coordination problem. So and Li proposed GA [8] algorithm to compute the optimal setting of DOCR. A modified PSO [9] was proposed by Zeineldin et al with less computation time to determine global optimum value of settings. Differential evolution (DE) is an efficient and powerful population-based stochastic search technique for solving optimization problems over continuous space, which has been widely applied in many scientific and engineering fields [10]. Bashir et al implemented a new hybrid PSO [11] to compute optimal coordination of DOCR considering TMS as continuous parameter and PS as discrete parameter. This paper is organized as follows: In Section 2, mathematical model formulation for solving the coordination problem is discussed. Section 3 describes about the enhanced differential evolutionary algorithm which have been applied to solve the problem. In Section 4, the proposed method is carried in 8 bus test system and the results are discussed. Finally, conclusions are summarized in Section 5.

2 Problem Formulation Identifying the relay requirements precisely and unambiguously and also the relay coordination criteria to be adopted are the most important and primary task to be undertaken by the protection engineers to ensure the high efficiency of system protection. Relay characteristics and coordination criteria are summarized below. 2.1 Relay Characteristics The operating time of the overcurrent relay is a non-linear function of pickup current setting (Ip) and time dial setting (TDS). The pickup current is the minimum current value above which the relay trips. The time dial setting determines the operation time of the relay for each current value given by the typical time characteristic T Vs M. Where, M (multiple of the pickup current) is the ratio of the relay fault current to the pickup current setting, i.e. M=If/Ip. The relay characteristics function can be represented by equation (1) and is given by equation (2). Ti=fi(TDSi,Ipi,Ifi) T=

(1)

( 0.14*TDS) (M 0.02 − 1 )

(2) Where, TDSi is the time dial setting of relay Ri, Ipi is the pickup current setting of relay Ri and Ifi is the current flowing in the relay Ri for a particular fault located in zone k. 2.2

Relay Setting

The calculation of the two settings, TDS (discrete) and IP (discrete) is the most important part of the overcurrent relay coordination. IP also correspond to the plugsetting of the relay. Each relay pickup current has lower and upper limit values. The

Enhanced Discrete Differential Evolution to Determine Optimal Coordination

87

lower limit is the larger of the minimum tap available or a factor times the maximum load current flowing through the relay. The upper limit is the smaller of the fault current. Similarly TDS has also lower and upper limit values based on the relay current-time characteristic. The constraints can be formulated as

I pi min < I pi < I pi max

(3)

TDS i min < TDS i < TDS i max

(4)

2.3 Coordination Criteria \To ensure relay coordination, the operating time of the backup relay should exceed that of its corresponding primary relay for all the faults by a coordination time interval (CTI). The typical value of CTI is 0.2 sec. CTI constraint is illustrated.

Tbackup − Tprimary ≥ CTI

(5)

Where, Tbackup is the operating time of backup relay and Tprimary is the operating time of the primary relay. After considering all these criteria, overcurrent relays coordination problem can be formulated as minimization of N

J =

T i =1

pri

(6) Where N is the number of relays operating and Tpri is the operating time of the primary relay for near-end fault.

3 Enhanced Differential Evolution Evolutionary Algorithms (EAs), inspired by the natural evolution of species, have been successfully applied to solve numerous optimization problems in diverse fields. The differential evolution (DE) algorithm, proposed by Storn and Price [12], is a simple yet powerful population-based stochastic search technique, which is an efficient and effective global optimizer in the continuous search domain. DE has been successfully applied in diverse fields such as power systems [13], mechanical engineering [13], communication [13] and pattern recognition [13]. In DE, there exist three crucial control parameters, i.e., population size, scaling factor, and crossover rate, may significantly influence the optimization performance of the DE. DE algorithm aims at evolving a population of NP D–dimensional parameter vectors, so-called individuals, which encode the candidate solutions towards the global optimum. The parameters of the ith vector for the generation g are given by equation (7).

{

X (gi ) = x(gi ,1) , x(gi , 2 ) ,  , x(gi , j ) ,  , x(gi , D )

}

(7)

g = 1,2, ... , G and i = 1,2, ... , N, where G is the maximum number of generations, N is the population size and D is the dimension of the problem. The initial population

88

J. Moirangthem et al.

should better cover the entire search space as much as possible by uniformly randomizing individuals within the search space constrained by the prescribed minimum and maximum parameter bounds.

{

}

(8)

{

}

(9)

min X min = x1min , x2min,, xmin j ,, xD max X max = x1max , x2max ,, x max j ,, xD

The jth parameter of the ith vector at the first generation is initialized randomly using the equation

x (1i , j ) = x min + rand j

(

× x max − x min j j

(i, j )

)

(10)

Where rand(i,j) represents a uniformly distributed random variable within the range (0,1). 3.1 Mutation Operation After initialization, DE employs the mutation operation to produce a mutant vector V with respect to each individual X, so-called target vector, in the current population. For each target vector X at the generation g, its associated mutant vector is can be express as,

{

V( ig) = v(gi ,1) , v(gi , 2) ,  , v(gi , j ) ,  , v(gi , D )

}

(11)

It can be generated via certain mutation strategy. For example, the five most frequently used mutation strategies implemented in the DE are listed as follows: 1) “DE/rand/1”:

(

)

(

)

V(ig) = X (gα ) + F × X (gβ ) − X (gγ ) 2) “DE/best/1”:

g V(ig) = X best + F × X (gα ) − X (gβ )

(12)

(13)

3) “DE/rand-to-best/1”:

(

)

(

(

)

(

)

(15)

(

)

(

)

(16)

g V(ig) = X(gi) + F1 × Xbest − X(gi) + F2 × X(gα) − X(gβ )

)

(14)

4) “DE/best/2”: g V(ig) = X best + F1 × X (gα ) − X (gβ ) + F2 × X (gγ ) − X (gζ )

5) “DE/rand/2”:

V(ig) = X (gα ) + F1 × X (gβ ) − X (gγ ) + F2 × X (gζ ) − X (gη )

Enhanced Discrete Differential Evolution to Determine Optimal Coordination

89

The indices α, β, γ, ζ, η are mutually exclusive integers randomly generated within the range [1, N], which are also different from the index i. These indices are randomly generated once for each mutant vector for a particular generation g. The scaling factor F, F1 and F2 are positive control parameters for scaling the difference vector. X(best) is the best individual vector with the best fitness value in the population at generation g. 3.2 Crossover Operation After the mutation phase in the generation g, crossover operation defined by equation (18) is used to generate each trial vector represented in equation (17).

{

U (gi ) = u (gi ,1) , u (gi , 2 ) ,  , u (gi , j ) ,  , u (gi , D )

 v(gi , j ) u(gi , j ) =  g  x(i , j )

if (rand (i , j ) ≤ Cr ) or otherwise

}

(17)

j = jrand   

(18)

The crossover rate Cr is a user-specified constant within the range (0, 1), which controls the fraction of parameter values copied from the mutant vector. jrand is a randomly chosen integer in the range [1, D]. The binomial crossover operator copies the jth parameter of the mutant vector V(i) to the corresponding element in the trial vector U(i) if rand(i,j) ≤ Cr or j=jrand . Otherwise, it is copied from the corresponding target vector X(i). This ensures that the trial vector will differ from its corresponding target vector by at least one parameter. 3.3 Selection Operation After the objective function values of all trial vectors are evaluated, a selection operation is performed. The objective function value of each trial vector f(U(i)) is compared to that of its corresponding target vector f(X(i)) in the current population. If the trial vector has less or equal objective function value than the corresponding target vector, the trial vector will replace the target vector and enter the population of the next generation. Otherwise, the target vector will remain in the population for the next generation. The selection operation can be expressed as follows:

X

g +1 (i )

U (gi) = g  X (i )

if if

( ) ( ) ( ) ( )

f U (gi) ≤ f X (gi)   f U (gi) > f X (gi) 

(19)

3.4 Enhanced Discrete DE The differential evolution used in this paper requires operation on discrete values since the decision variables to be optimized (TDS and Ip) are discrete by nature. The input dimensions of the objective function to be optimized (summation of the operating time of all the relays), possess piecewise existential domains. So, before evaluation of the each potential solution vector in DE, the vectors generated are quantized to two digit precision when expressed in engineering notation.

90

J. Moirangthem et al.

The DE algorithm is enhanced by making the scaling factors F, F1 and F2 time varying and stochastic using the equations (20)-(22).

F g = F* + λ ⋅φ g

(20)

F1g = F1* + λ1 ⋅ φ1g

(21)

F2g = F2* + λ2 ⋅ φ2g

(22)

Where Fg values are the time varying enhanced scaling factors F* values are the constant offsets which act as the mean around which the scaling factors oscillate. The φ values represent random values which follow normal distribution function with zero as mean. The λ values are the multipliers which amplify the random values. Since the Fg values act as multipliers of differential vectors, it is often possible for the DE to get stuck at a local optimum. As the domains of the decision variables are discontinuous, it is difficult for the conventional algorithm to perturb the potential solutions so that the decision variables can have values which exist in their feasible domain. The time varying scaling factors ensure that the mutation operation causes necessary variation in the decision variables so that they can jump to other feasible points. This makes the DE enhanced for such a discrete problem.

4 Simulation Results 4.1 System Data The proposed method for solving the coordination problem was tested on 8 bus test system which is shown in Fig.1. The TDS are assumed to vary from 0.1 to 1.1 and the available pickup current setting is [0.5, 0.6, 0.8, 1, 1.5, and 2.5]. The coordination time interval (CTI) is taken as 0.2.

Fig. 1. 8 bus test system

Enhanced Discrete Differential Evolution to Determine Optimal Coordination

91

The three phase faults are applied at the near-end of each phase relay. The CT ratios for the pickup current setting for 3 bus system are given Table 1. The primary and back up pairs and also the fault currents are given in Table 2. Table 1. CT ratio for 8 bus system Relay No. 1 2 3 4 5

CT Ratio

Relay No. 6 7 8 9 10

1200/5 1200/5 800/5 1200/5 1200/5

CT Ratio 1200/5 800/5 1200/5 800/5 1200/5

Relay No. 11 12 13 14

CT Ratio 1200/5 1200/5 1200/5 800/5

Table 2. Primary and Backup pair information for 8 bus systems

Relay No. 13 3 4 12 11 8 11 2 1 9

Primary Fault current (If), Amps 1.400 0.176 2.700 1.700 0.831 0.202 0.202 0.666 0.745 1.700

Relay No. 1 2 3 4 5 6 7 8 9 10

Backup Fault current (If), Amps 5.000 4.900 4.300 4.300 4.800 4.900 5.300 5.000 4.900 4.300

Relay No. 10 6 5 1 14 14 12 7 7 2

Primary Fault current (If), Amps 2.600 0.485 1.600 2.600 0.666 1.800 0.485 0.086 1.800 0.086

Relay No. 11 12 13 14 2 6 7 8 12 14

Backup Fault current (If), Amps 4.300 4.800 4.900 5.300 4.900 4.900 5.300 5.000 4.800 5.300

4.2 Implementation of Enhanced Discrete Differential Evolution DE applied to the relay coordination problem formulation and was coded in MATLAB with total number of variables 28 and population size of 20 particles. The maximum number of generation count used is 1000. The enhanced discrete DE * * performs best with the last strategy in DE. The scaling factor F1 and F2 are taken to be as 0.5 and 0.3. Both λ1 and λ2 were taken as 0.8. The normal distribution function

φg

φg

(represented by 1 and 2 ) generates positive and negative values in continuous domain with mean as zero. The crossover rate used is 0.8. The optimal time dial setting and pick up current setting of each overcurrent relay for the test systems are calculated. The optimal values are shown in Table 3. The results obtained by using Enhanced Discrete DE are compared with other algorithms and found that the Enhanced DE algorithm gave better performance than other algorithms. Performance of various comparable algorithms in terms of optimal operating time is given in table 4.

92

J. Moirangthem et al. Table 3. Optimal setting of relays for 8 bus test system Relay No. TSM Ip 1 0.1 0.5 2 0.1 0.5 3 0.1 0.5 4 0.1 0.5 5 0.1 0.8 6 0.1 0.5 7 0.1 0.5 Objective function value (J)

Relay No. 8 9 10 11 12 13 14

TSM 0.1 0.1 0.1 0.1 0.1 0.1 0.1 2.6245

Ip 0.5 1.5 0.5 0.5 0.5 0.5 0.5

Table 4. Comparisons of optimal setting of relays with various algorithms Algorithm Genetic Algorithm (GA)

Objective function values (J) 2.7620 s

Particle Swarm Optimization (PSO)

3.6272 s

Enhanced Discrete Differential Evolution (EDDA)

2.6245 s

5 Conclusion The details of the proposed algorithm and program implementation for directional overcurrent relay coordination have been discussed in this paper. The problem has been formulated as non-linear problem taking into account both the parameter i.e., TDS and Ip as discrete. Computer results for the 8 bus test system have been reported and compared with different algorithms. The obtained results show that the proposed algorithm (EDDEA) gave better optimal operating time of relay without miscoordination when compared to other algorithms.

References [1] Ramaswami, R.: Transmission Protective Relay Coordination- A Computer-AidedEngineering Approach for Subsystems and Full Systems, Ph.D. Dissertation, University of Washington Seattle (1986) [2] Urdenta, A., Restrepo, H., Marques, S., Sanches, J.: Optimal Co-ordination of directional overcurrent relay timing using linear programming. IEEE Trans. Power Delivery 11, 122–129 (1996) [3] Urdenta, A., Nadira, R., Prez, L.: Optimal Co-ordination of directional overcurrent relay in interconnected power system. IEEE Trans. Power Delivery 3, 903–911 (1988) [4] Braga, A.S., Saraiva, J.T.: Co-ordination of directional overcurrent relays in meshed networks using simplex method. In: Proc. IEEE MELECON Conf., vol. 3, pp. 1535–1538 (1996) [5] Abyaneh, H.A., Keyhani, R.: Optimal co-ordination of overcurrent relays in power system by dual simplex method. In: Proc. AUPEC Conf., Perth, Australia, vol. 3, pp. 440–445 (1995)

Enhanced Discrete Differential Evolution to Determine Optimal Coordination

93

[6] Birla, D., Maheshwari, R.P., Gupta, H.O.: Time overcurrent relay coordination: a review. International Journal of Emerging Electric Power Systems 2(2), 1–13 (2005) [7] Birla, D., Maheshwari, R.P., Gupta, H.O., Deep, K., Thakur, M.: Application of random search technique in overcurrent relay coordination. International Journal of Emerging Electric Power Systems 7(1), 1–14 (2006) [8] So, C.W., Li, K.K., Lai, K.T., Fung, K.Y.: Application of genetic algorithm for overcurrent relay coordination. In: Proc. IEE Conference on Developments in Power System Protection, pp. 66–69 (1997) [9] Zeineldin, H.H., El-Saadany, E.F., Salama, M.M.A.: Optimal coordination of overcurrent relays using a modified particle swarm optimization. Electrical Power System Research 76, 988–995 (2006) [10] Qin, A.K., Huang, V.L., Suganthan, P.N.: Differential Evolution Algorithm with Strategy Adaptation for Global Numerical Optimization. IEEE Trans. Evolutionary Computation 13, 398–417 (2009) [11] Bashir, M., Taghizadeh, M., Sadeh, J., Mashhadi, H.R.: A new hybrid particle swarm optimization for optimal coordination of overcurrent relay. In: International Conference on Power System Technology, pp. 1–6 (2010) [12] Storn, R.: Differential evolution: A simple and efficient adaptive scheme for global optimization over continuous spaces, ICSI, Tech. Rep. TR-95-012 (1995) [13] Das, S., Suganthan, P.N.: Differential Evolution: A Survey of the State-of-the-Art. IEEE Trans. Evolutionary Computation 15(1), 4–31 (2010)

Dynamic Thinning of Antenna Array Using Differential Evolution Algorithm Ratul Majumdar, Aveek Kumar Das, and Swagatam Das Jadavpur University, Kolkata-700032, India {majumdar.ratul,aveek23}@gmail.com, [email protected]

Abstract. Dynamic thinning is a process by which the total number of active elements in an antenna array is reduced under real time conditions without causing major degradation in system’s performances. This paper suggests a technique by which a thinned antenna array is designed by simple differential evolution(DE) algorithm overcoming the difficulties of large and rugged solution space and slow rate of convergence in case of large 2-D arrays. Keywords: Antenna Array,Dynamic thinning, Differential Evolution, Zoning Technique.

1 Introduction Thinning refers to strategic elimination of a subset of active elements within an antenna array, keeping the deviation of the resulting radiation pattern within limits. All the results present in this paper are based on study of Array factor of a set of isotropic radiators placed at a uniform spacing of half wavelength. The array factor of a linear array of M identical antenna elements is given by,

FM (θ ) = i =1 Ri e jkd i cos(θ ) M

(1)

Ri is the complex excitation coefficient of the i th element located at x = d i along the linear array direction x and k is the wave number. Now to obtain a new linear array that has minimum number of elements which differs from, FM (θ ) by where

less than a prescribed tolerance of ε , with all other conditions remaining the same. So, we have to find a solution to,

 min F (θ ) − Q R e jkdi cos(θ )  ≤ ε { R ,d }  i=1 i M  i i i =1,,Q 

(2)

Ri and d i (i = 1,..., Q ≤ M ) are the complex excitations and locations for Q antenna elements respectively, for all values of θ . where

B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 94–101, 2011. © Springer-Verlag Berlin Heidelberg 2011

Dynamic Thinning of Antenna Array Using Differential Evolution Algorithm

95

2 Objective Function Construction of a good objective function is an important issue in any antenna array design problem. In this paper the following objective function was used for the thinned antenna array synthesis: 2

n) ε (θ ) = [1 + ω1 θ 0 − θ m( n ) (ν ) ].SLL(max (ν ).[1 + ω 2U ( SLLmax (ν ) − SLL0 )]

where ( n) is the number of evolution generations, ω i (i

(3)

= 1,2) is the weighing factor

for the different terms to the objective function. SLLmax (ν ) is the calculated (n)

maximum SLL and

θ o and θ M are

desired and calculated main beam directions

respectively. In equation (3), U is a step function used for restriction of for prescribed

n) SLL(max (ν )

n) SLL0 such that U = 1 for SLL(max (ν ) > SLL0 and U = 0 for

n) SLL(max (ν ) ≤ SLL0 .This objective function is found to be ideal for using with

simple differential evolution (DE) algorithm.

3 Overview of Differential Evolution Differential evolution algorithm, proposed by Storn and Price [1] in 1995, is a simple but effective population-based stochastic search technique for solving global optimization problems. The algorithm is named differential evolution (DE) owing to a special kind of differential operator, which they invoked to create new off-spring from parent chromosomes instead of classical crossover or mutation. Let S ⊂ ℜ be the D dimensional search space of the problem under construction. The D.E. evolves a population of NP, D-dimensional individual vectors, n

 X i ,G = (xi ,1 , xi , 2 ,  , xi ,D

) ∈S , i = 1,2  , NP

from one generation to another. The

primary features of D.E can be stated as follows [1, 2]: 3.1 Initialization The 1st step of D.E. Algorithm is the initialization of the population. The population should ideally cover the entire search space between the prescribed upper







= {x1,max , x2,max ,......, x D ,max }  and X min = {x1,min , x2 ,min ,.......x D ,min } . The jth component of the ith individual is

and

lower

bounds X max and X min where X max

initialized as follows:

xi , j = x j ,min + rand i j (0,1) ( x j ,max − x j ,min ) ; j ∈ [1,D] j

Here rand i (0,1) is a uniformly distributed random number within [0, 1].

(4)

96

R. Majumdar, A.K. Das, and S. Das

3.2 Mutation



For every target vector, X i ,G , in any generation G, a mutant vector

 Vi ,G is

generated. The most common mutation policies generally used in DE are:









1. DE/rand/1/bin: V i ,G = X r1,G + F .( X r1,G − X r 3,G )

    2. DE/best/1/bin: Vi ,G = X best,G + F ⋅ ( X r1,G − X r 2,G )       3. DE/target-to-best/1/bin: Vi,G = X i,G + F.(X best,G − X i,G ) + F1.( X r1,G − X r 2,G )

(5) (6) (7)

where r1, r2, and r3 are random and mutually exclusive integers generated in the range [1, NP], which should also be different from the trial vector’s current index i. F  and F1 are weight factors in [0, 2] for scaling differential vectors and X best ,G is the individual vector with best fitness value in the population at generation G. 3.3 Crossover This operation involves binary crossover between the target vector mutant vector

 X i ,G and the

 Vi ,G produced in the previous step which produces the trial vector

 U i ,G = (ui ,1, G , ui , 2, G ,  , ui , n , G ) . The crossover operation is done as follows. v ui, j,G =  i,j,G xi,j,G

if rand(0 ,1 ) ≤ CR or j = jrand

(8)

otherwise

where CR is a user-specified crossover constant in the range [0, 1) and jrand is a randomly chosen integer in the range [1, NP] to ensure that the trial vector differ from its corresponding target vector

 X i ,G by at least one parameter.

 U i ,G will

3.4 Selection If the values of some parameters of a newly generated trial vector exceed the corresponding upper and lower bounds, we randomly and uniformly reinitialize it



within the search range. The fitness value of each trial vector f ( U i ,G ) is compared to



that of its corresponding target vector f ( X i ,G ) in the current population and the population for the G+1 generation is formed as follows: (for a minimization problem)    if f (Ui,G ) ≤ f ( X i,G ) Ui,G  X i ,G+1 =   otherwise X i,G (9)

where f (.) is the objective function.

Dynamic Thinning of Antenna Array Using Differential Evolution Algorithm

Start

Initialize the population Gen=1 Evaluate the population using the fitness function f

Create off-springs and evaluate their fitness

Is fitness of off-springs better than fitness of parents?

Gen=Gen+1 Yes

No Discard the offsprings in the new population

Replace the parents by off springs in the new population No Gen>Max_Gen?

Yes Stop

Fig. 1. The Flowchart describes the Differential Evolution Operation

97

98

R. Majumdar, A.K. Das, and S. Das

4 Application of Differential Evolution to Dynamic Array Thinning It can be shown that the Differential Evolution (DE) algorithm is successful in designing a thinned array which can be accomplished by strategically removing a subset of active elements in the array to achieve the objective function. Now, this type of problem can be represented in a binary format by representing the active elements as value of ‘1’ and inactive elements are represented by value of ‘0’. So the problem reduces to finding the appropriate pattern of ‘1’s and ‘0’s in order to satisfy the constraints. The first step of applying DE to the above is to initialize a set of random binary strings as the initial population where a ‘1’ represents an active element and a ‘0’ represents an inactive element in a particular string. Then the basic steps of DE involving mutation, recombination and selection is done iteratively on the population until end criteria is met. The Side Lobe Level (SLL) so obtained is compared with that of Simple Genetic Algorithm (SGA).It is seen from fig.2 that for a 100 element uniformly excited antenna array compared to the -20.5dB peak SLL of thinned antenna array by SGA the peak SLL by DE algorithm is -23dB.In both the cases a total of 20 array elements are found to be inactive.

Fig. 2. Radiation pattern of a uniformly excited array after thinning using DE; Due to symmetry only half the pattern is shown

4.1 Dynamic Array Thinning Dynamic array thinning is a process by which thinning pattern is varied on a real time basis to adapt to varying conditions. Although it is seen that performance of DE algorithm is better than SGA in solving thinned antenna arrays it still suffers the

Dynamic Thinning of Antenna Array Using Differential Evolution Algorithm

99

problem of computational complexity and finding an ‘acceptable solution’ for the real world scenario. 1. Computational Complexity: The evaluation of objective function usually involves long procedures. Also, for linear 2-D antenna arrays as the number of antenna elements increases the solution space within which a required solution can be found also increases. As a result exploring such a large solution space would not only require time but there is a chance of premature convergence. 2. Concept of Acceptable Solution: Generally an optimization problem would strive at getting at globally optimized solution within the set constraints. However, in a realtime scenario as in the case of dynamic thinning, a solution which gives a nearoptimum value would be more useful input than the absolute minimum value. So, dynamic thinning should look for acceptable solution rather than the absolute minimum value. The computational complexity and the problem of acceptable solution can be taken care of by the zoning technique [3] which is described below: 4.2 Zoning Technique Zoning is the process by which the antenna array is partitioned into a number of convenient zones N Z so that the solution space can be explored usefully. It is assumed that each zones consists of a minimum of two elements and N Z =1 represents an undivided antenna array. It is evident that the number of inactive elements in the central portion of the linear array is generally less than the other portions of the array. Such an assumption helps in better exploitation of the solution space. Thus, Zoning can help in enhancing convergence rate, since the proportion of the exploring space to total solution space reduces drastically in case of large arrays.

5 Parameter Values Used for the Experiment The parameters for the Differential Evolution algorithm in this particular experiment performed are: 1.

Weight Factor: F and F1 are both taken as F= F1= F_weight + rand (0, 1)*F_weight; F_weight =0.5

2. 3.

(10)

Crossover Constant: The value of the Crossover constant (CR) is taken as 0.7 Mutation Scheme of DE: The mutation scheme in the DE part of the algorithm applied to the superior sub-population is the “DE/target-to-best/1/bin” scheme that follows the mutation according to equation (7)

100

R. Majumdar, A.K. Das, and S. Das

6 Experimental Results The simulation was carried on symmetrical arrays with isotropic radiators placed at a uniform spacing of half wavelength. Study was mainly carried out to see the effect of the number of zones on the speed of convergence. The Zoning details of 200 element linear antenna array are shown in table1. The objective function was chosen to achieve -24dB SLL outside the sector of ± 0.01 radians. Maximum number of iterations taken is N it = 1000 and the termination criterion is to achieve either the SLL or the maximum number of iterations whichever was earlier. The average of 50 runs of the said algorithm is shown below. The convergence behavior of DE was compared with SGA [3] after observing (i)Number of runs in which objective function was achieved within N it . (ii)Number of iterations required to achieve the desired SLL averaging over 50 runs. (iii)SLL actually obtained while termination criteria is satisfied averaging over 50 runs. Figure 3 shows the convergence trend of normalized SLL for the four zoning approaches used. For each curve, at every iteration, an average value of normalized SLL over 50 runs is calculated. Table 1. Comparative Study of the SLL for DE and SGA algorithms

Nz =1

Nz = 2

Nz = 3

Nz = 4

Zone1:100

Zone 1: 40 Zone 2: 60

Zone 1: 40 Zone 2: 30 Zone 3: 30

Zone 1: 40 Zone 2: 20 Zone 3: 20 Zone 4: 20

1

80

20

3

1

90

40

3

DE

457

346

358

330

SGA

496

337

382

448

DE

-21.36

-23.15

-22.42

-23.68

SGA

-20.35

-22.13

-22.03

-21.84

Algorithm used Zoning details (Number of elements in half array) Number of runs in which desired SLL was achieved within

N it

Number of iterations to achieve desired SLL averaged over 50 runs SLL (dB) obtained while termination, averaged over 50 runs

DE SGA

Dynamic Thinning of Antenna Array Using Differential Evolution Algorithm

101

1 Nz=1 Nz=2 Nz=3 Nz=4

0.9 0.8

Norm alis ed S LL

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

50

100

150

200 250 300 Number of Iterations

350

400

450

500

Fig. 3. Convergence curves for study of the number of zeroes

From the figure it is evident that best convergence is achieved for Nz=4.

7 Conclusion Dynamic array thinning is a very important concept for real time thinning of antenna arrays in varying conditions. Simple differential evolution (DE) algorithm provides an easy and effective way to achieve this. However, for large 2-D arrays Zoning technique is employed for computational complexity and to solve the problem of ‘acceptable solution’. The simulation results thus found were much better than simple genetic algorithm (SGA).

References 1. Storn, R., Price, K.: Differential Evolution–A Simple and Efficient Heuristic for Global Optimization Over Continuous Spaces. Journal of Global Optimization 11(4), 341–359 (1997) 2. Das, S., Abraham, A., Konar, A.: Particle Swarm Optimization and Differential Evolution Algorithms: technical analysis, Applications and Hybridization Perspectives, http://www.softcomputing.net/aciis.pdf 3. Jain, R., Mani, G.S.: Dynamic Thinning Of Antenna Array Using Genetic Algorithm. Progress In Electromagnetics Research B 32, 1–20 (2011) 4. Bray, M.G., Werner, D.H., Boeringer, D.W., Machuga, D.W.: Optimization of thinned aperiodic linear phased arrays using genetic algorithms to reduce grating lobes during scanning. IEEE Trans. on Antennas and Propagation 50(12), 1732–1742 (2002) 5. Mahanti, G.K., Pathak, N., Mahanti, P.: Synthesis of Thinned Linear Antenna Arrays with Fixed Side lobe level using Real-Coded Genetic Algorithm. Progress In Electromagnetics Research, PIER 75, 319–328

A Quantized Invasive Weed Optimization Based Antenna Array Synthesis with Digital Phase Control Ratul Majumdar, Ankur Ghosh, Souvik Raha, Koushik Laha, and Swagatam Das Dept. of Electronics & Telecommunication Engg , Jadavpur University, Kolkata-700 032, India {majumdar.ratul,ankurg708,rahasouvik37,lahakoushik1}@gmail.com, [email protected]

Abstract. The design of antenna arrays aims at minimizing side-lobe levels far as practicable. So a major concern of designer is to optimize the side lobes to increase directivity, gain, and efficiency. Invasive Weed Optimization (IWO) is a recently developed, ecologically inspired metaheuristic algorithm that has already found some major applications in electromagnetic research. In this article the synthesis of array antenna pattern by digital phase shifters is accomplished with a modified version of the IWO algorithm called QIWO (Quantized Invasive Weed Optimization. The proposed algorithm searches for optimal solution of the fitness function, which contains the SLL value and the interference suppression keeping the main beam unchanged. The results obtained from this algorithm are better than that of QPSO (Quantized Particle Swarm Optimization) and BPSO (Binary Particle Swarm Optimization). In this paper the array factor is expressed mathematically by a linear transform, which is similar to Discrete Cosine Transform (DCT). This proposed algorithm is also found to be efficient in computing for large arrays.

1 Introduction In an array of identical elements, there are five controls that can be used to shape the overall pattern of the antenna array. These are (1) the geometrical configuration of overall array (linear, circular, rectangular, spherical etc.) (2) relative displacement between the elements, (3) excitation amplitude of individual elements (4) excitation phase of individual elements and (5) the relative pattern of individual elements. The theory related to this topic can be found in [1]-[3]. Among these the main factor of concern here is the excitation phase of individual array elements as these networks can minimize excitation errors and preserve coherence. In recent times, use of digital phase shifters has become very popular for beam scanning and interference suppression. Usually, the optimized phase solutions are calculated assuming the phase shifters to be variables that are continuous not discrete.. Thus, the actual calculated phases must be rounded to the nearest quantized value of digital phase shifters for simplification and also this is transferred into digital domain. Recently Mehrabian and Lucas proposed the Invasive Weed Optimization (IWO) [4], a metaheuristic algorithm, mimicking the ecological behavior of colonizing weeds. Here we use a modified version of IWO. For the particular problem we restrict the solution space B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 102–109, 2011. © Springer-Verlag Berlin Heidelberg 2011

A Quantized Invasive Weed Optimization Based Antenna Array Synthesis

103

into finite discrete phase values. So the proposed Quantized IWO (QIWO) will search for an optimal solution within the available quantized values for digital phase shifters. Here the Side Lobe Levels (SLLs) are optimized for designing the antenna array [5] and the objective function includes a weighted sum of the mean beam width value, the sum of the nulls’ depths at the interference signals directions, and the SLL value. To allow high speed solution of the objective or fitness functions, the array factor is expressed linearly, based on the Discrete Cosine Transform with pre-computed DCT matrix The proposed method provides significant performance improvement over Quantized Particle Swarm Optimization (QPSO) [6] and Binary Particle Swarm optimization (BPSO) [7, 8].

2 Digital Phase Antenna Array Figure 1 shows a uniformly excited Centro-symmetric linear array of 2N isotropic equi-distant elements with inter element spacing d. Each element of the array is followed by a digital phase shifter ϕ . Thus the array factor with main beam steering is expressed as,

F (θ ) = N

−1

 a ( m ).e

j .m .d . k  cos (θ ) − cos (θ s ) 

+

m =− N

 a (n) e

j .n .d .k  cos (θ ) − cos (θ s ) 

,

n =1

(1) where

a (n) is equal to

e

jϕ n

, k is the wave number, θ is the scanning angle, and

θ s is the main beam angle. The first term corresponds to the array factor of the left hand half of the array, and the second term corresponds to the array factor of the right hand half of the array. By letting m = − n the array factor can be written as:

F (θ ) = = F1 (π

1

N

n= N

n =1

 a(n)e − jndk [cos(θ )−cos(θ s )] +  a(n )e jndk [cos(θ )−cos(θ s )]

− θ ) + F2 (θ )

If the phase shifter values of the array

(2)

ϕn

are taken in a symmetrical manner relative to the center

a( n) = a(− n) then the Centro-symmetric array of size 2N is reduced to, N

(

F (θ ) = 2 a ( n ) cos n.d .k . ( cos (θ ) − cos (θ s ) ) n =1

)

(3)

For one part of the array the arrival angle is θ while for the other part of the array the arrival angle is π − θ . Now we discretize (3) with M points and the above equation can be written in matrix form by multiplication of two matrices having sampled

104

R. Majumdar et al.

values of a(n), & W( θ i , n ) with,

W (θ i , n) = cos(ndk (cos(θ i ) − cos(θ s ))) . Clearly the product of matrices containing the sampled values of a(n) and W( θ i , n ) gives the same result as obtained by its continuous equivalent i.e. eqn. (3):

W (θ 1 ,1)  W (θ 1 , N )   F (θ 1 )      = 2        W (θ M ,1)  W (θ M , N )  F (θ N )

a(1)      a( N ) (4)

The above transformation is similar to performing discrete cosine transform.

Fig. 1. Centro-symmetric antenna array using digital phase shifters

3 IWO and Its Proposed Modification IWO is a population-based algorithm based on trial and error method that copies the colonizing behavior of weeds. Weed grows its population entirely or in a specified area without any control of any external factor in a very random manner. Initially a certain number of weeds are randomly spread over the entire search range. These weeds will eventually grow up and execute the following steps and the algorithm proceeds. 3.1 Classical IWO There are four steps in the classical IWO algorithm as described below: Initialization. A finite number of weeds are initialized randomly in the search space.

A Quantized Invasive Weed Optimization Based Antenna Array Synthesis

105

Reproduction. Each member of the population is allowed to produce seeds depending on its own, as well as the colony’s lowest and highest fitness, such that, the number of seeds produced by a weed increases linearly from lowest possible seed for a weed with the worst fitness to the maximum number of seeds for a weed with the best fitness. Spatial distribution. The generated seeds are randomly scattered over the ddimensional search space by perturbing them with normally distributed random numbers with zero mean and a variable variance. However, the standard deviation (sd) of the random function is made to decrease over the iterations. If sd max and sd min are the maximum and minimum standard deviations and if pow is a real number, then the standard deviation for a particular iteration can be given as in eqn. (5):

sd ITER

 iter − iter  =  max   itermax 

pow

( sd max − sd min )

+ sd min ,

(5)

Where iter is the current iteration number and itermax equals the maximum number of iterations allowed. Competitive Exclusion. Some kind of competition between plants is needed for limiting maximum number of plants in a colony. Initially, the plants in a colony will reproduce until the number of plants reaches a maximum value pop max . From then on, only the fittest plants up to pop_max, among the existing ones and the reproduced ones, are taken in the colony and steps 2 to 4 are repeated until the maximum number of iterations has been reached, i.e. the colony size is fixed from thereon to popmax . This method is known as competitive exclusion and is also a selection procedure of IWO. 3.2 The Quantized IWO The algorithm described above is based on continuous variables which are difficult to perceive in this context of digital phase shifters. This forced us to restrict the solution space into some quantized values of digital phase shifters consequently; QIWO will search for optimal solution within the available quantized values. The algorithm starts by randomly initializing integer particle values. Each value or level l has the ability to change to a higher value l+1 or to a lower value l-1.Any value exceeding the maximum level L-1 is set to L-1, and any negative value is set to the lowest level 0.After bounding the positions and quantization is performed the fitness of each solution is obtained and the best solution is found by IWO. In our optimization problem, a plant is the set of digital values of the phase shifters associated with each element of the half of the antenna array of size N whose position is given in N dimensional space by X q = ϕ q = ψ q1 ,ψ q 2 ,.....,ψ qN where ϕ q is the

[

]

phase of an element of the antenna array. Each phase can have any value in the

106

R. Majumdar et al.

discrete space [0…15] since the total phase has been quantized in 16 divisions and therefore can be written with hexadecimal notation [0…F] which is encoded using 4 bits. Quantization means discrete amplitude representation of a physical quantity. .We contemplate the original physical quantity whose excursion is confined to a certain range ,the total range being divided into equal intervals .In the center of each of these intervals we locate quantization levels. At every instant of time, the approximated version of the physical quantity takes the value of the quantization level to which the recent value of the original physical quantity is closest. A: minimum SLL optimization:

F1 = min( AF (θ )),

(6)

Where array factor is a function of θ B: optimization with interference suppression:

F2 = C1 + C2 , C1 = K ( SLLO − SLLd ) 2

(8)

C 2 = θ = −90o [W (θ ) AF0 (θ ) − AFd (θ ) ]

(9)

Where And

(7)

θ = +90 0

The first term of the cost function in (7) is used for the purpose of SLL suppression. SLL0 and SLLd in (8) are respectively the normalized side lobe level of actual pattern and the desired pattern. Coefficient k is the weight, controlling the optimization process. This second term of the cost function in (7) is for interference suppression i.e. to place nulls at specified directions, where, AF0 (θ) is the pattern obtained by using QIWO, AFd (θ) is the desired pattern, and W (θ) is the controlling parameter for creating the null. Now we have applied the proposed QIWO algorithm on the final objective function F3 (θ ) defined as:

F3 (θ ) = μ1F1 (θ ) + μ 2 F2 (θ ) , Where μ1 and

μ2

(10)

are the weights signifying the importance of the individual terms.

4 Experiments and Results The computing was achieved on a Pentium 4 computer running at 3.0 GHz equipped with 1.5 GB of RAM. Results are presented for arrays of 100 and 200 elements for a Centro-symmetric antenna. The only controlled parameter of the array is the element digital phase shifter which is 4 bit in all the following results.. In general, the values of the phases can have any discrete values between –П and + П .Running a large number of computer simulation examples we find that that a 4 bit phase shifter is a good choice with shift angles range set from 0º to 120º with 6.50º increment step. In term of computation time and based on the DCT implementation the search of optimum solutions among a space of 50,000 solutions is performed in only 15 seconds for an array size of 100 elements. This timing shows the efficiency of our implementation compared to QPSO algorithm.

A Quantized Invasive Weed Optimization Based Antenna Array Synthesis

107

4.1 Minimum SLL Optimization First ,an array factor is designed with low side lobe level by optimizing the digital phase shifters. The optimization algorithm takes into consideration two linear equispaced isotropic antenna arrays with 100 array elements (N=50) and 200 array elements (N=100) respectively. Figures 4 and 5 show the array factor with SLL equal to 21 dB and 25 dB respectively while the main beam width remains equal to the uniformly excited antenna array. The values of SLL obtained is compared to the values obtained by QPSO in [8] and shown in Table I. An optimum set of discrete solutions of the digital phase shifters in provided in TABLE II. The array factor is taken every 0.5 degree. Table 1. Comparison of QIWO with BPSO and QPSO N

50 100

Values obtained by QPSO algorithm (dB) 18 21

Value obtained by BPSO algorithm (dB)

Value obtained by QIWO algorithm (dB)

17 20

21 25

Fig. 2. The optimized array factor with minimum SLL only, d=λ/2, swarm size=100,SLL=21dB Table 2. Different solutions given by their digital phase sequences and SLLs FIGURE 2 3

5

6

DIGITAL PHASE SEQUENCE 000FA45888BDAC490EDC372A8 5375AC2D7E46091267549530025 145372EDCA437693BCEF465842 487635EDCABF333289BDE3D6327 452579ACDE 46676561345286235884685EDACFB AECFF888003442414238642346242 48800ADDEE4235

SLL (dB) 21

25000428289438ADEF42242CED472 4256356DEACBD24563556AED425 EFACEEBACE

17

25

15

108

R. Majumdar et al.

Fig. 3. The optimized array factor with minimum SLL only, d =λ/2, swarm size=200, SLL=25Db

To prove the effectiveness of QIWO over QPSO we plot the SLL value versus iteration value for N=50 averaging 20 trials in Fig. 4.The ash plot and the red plot shows the SLL values for QIWO and QPSO respectively.

Fig. 4. SLL versus number of iterations using QPSO(red) and QIWO(black).the curves show the convergence by averaging 20 trials and taking N=50,swarm size=100.

Optimization with Interference Suppression. The digital phase shifters are used for narrowband and wideband interference suppression. The QIWO algorithm uses a swarm size of 100 and the fitness function given in (2).Fig. 5 shows two array factors with 15 dB SLL with one suppressed sectors of 1º starting from 120º compared to 12.4 dB SLL obtained from QPSO respectively. Also for the third fitness function we find that QIWO produces the best result for μ1 =0.6 & μ 2 =0.4.

A Quantized Invasive Weed Optimization Based Antenna Array Synthesis

109

Fig. 5. The optimized array pattern with one suppressed sector (120º-121º),d= λ/2,N=50

5 Conclusion The digital phase shifters are controlled for interference suppression while minimizing SLL by quantized invasive weed optimization (QIWO).The algorithm proposed here is more effective than QPSO because IWO in many applications gives better results than PSO and is faster than other algorithms. The particles with better fitness are allowed to reproduce and hence they find the optimum solution for the multimodal function more efficiently and quickly.

References 1. Baird, D., Rassweiler, G.: Adaptive sidelobe nulling using digitally controlled phase shifters. IEEE Trans. Antennas Propag. 24(5), 638–649 (1976) 2. Steyskal, H., Shore, R.A., Haupt, R.L.: Methods for null control and their effects on radiation pattern. IEEE Trans. Antennas Propag. 34, 404–409 (1986) 3. Haupt, R.L.: Phase-only adaptive nulling with a genetic algorithm. IEEE Trans. Antennas Propag. 45(6), 1009–1015 (1997) 4. Mehrabian, A.R., Lucas, C.: A novel numerical optimization algorithm inspired from weed colonization. Ecological Informatics 1, 355–366 (2006) 5. Oliveri, G., Poli, L.: Synthesis of monopulse sub-arrayed linear and planar array antennas with optimized sidelobes. Progress In Electromagnetics Research, PIER 99, 109–129 (2009) 6. Ismail, T.H., Hamici, Z.M.: Array pattern sunthesis using digital phase control by quantized particle swarm optimization. IEEE Transactions on Antennas and Propagation 58(6), 2142– 2145 (2010) 7. Jin, N., Rahmat-Samii, Y.: Advances in particle swarm optimization for antenna designs: Real-number, binary, single-objective and multiobjective implementations. IEEE Trans. Antennas Propag. 55(3), 556–567 (2007) 8. Kennedy, J., Eberhart, R.C.: A discrete binary version of the particle swarm algorithm. In: IEEE Int. Conf. Man Syst., vol. 5, pp. 4104–4108 (October 1997)

Optimal Power Flow for Indian 75 Bus System Using Differential Evolution Aveek Kumar Das1, Ratul Majumdar1, Bijaya Ketan Panigrahi2, and S. Surender Reddy2 1

Department of E.T.C.E, Jadavpur University, Kolkata Department of E.E, Indian Institute of Technology, Delhi {aveek23,majumdar.ratul,bijayaketan.panigrahi, salkuti.surenderreddy}@gmail.com 2

1

Introduction

The optimal power flow (OPF) has been commonly used as an efficient tool in the power system planning and operation for many years. OPF is an important tool in modern Energy Management System (EMS). It plays an important role in maintaining the economy of the power system. The problem of OPF subjected to set of equality and inequality constraints was originally formulated in 1962 by Carpentier [1]. The OPF problem is a nonlinear, non-convex, large scale, static optimization problem with both continuous (generator voltage magnitudes and active powers) and discrete (transformer taps and switchable shunt devices) control variables. Even in the absence of discrete control variables, the OPF problem [2] is non convex due to the existence of the nonlinear (AC) power flow equality constraints. The presence of discrete control variables, such as switchable shunt devices, transformer tap positions further complicates the problem solution. Evolutionary Algorithms (EAs) such as Genetic Algorithms (GA), Particle Swarm Optimization (PSO), Differential Evolution (DE) etc. have become the method of choice for optimization problems that are too complex to be solved using deterministic techniques such as linear programming or gradient (Jacobian) methods. EAs require little knowledge about the problem being solved, and they are easy to implement, robust and inherently parallel. Because of their universality, ease of implementation, and fitness for parallel computing, EAs often take less time [3] to find the optimal solution than gradient methods. OPF can be used periodically to determine the optimal settings of the control variables to minimize the generation cost, minimization of the transmission system losses. Genetic Algorithm (GA), Enhanced GA, Improved GA has been successfully applied for solution of OPF problem [4]-[6]. In [7], PSO was used to solve the OPF problem. Here, we propose the Differential Evolution algorithm, and a Hybridized Algorithm with D.E and I.W.O (with Population Exchange) which can handle both continuous and discrete control variables. The effectiveness of the differential evolution algorithm is tested on practical Indian 75 bus system data.

B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 110–118, 2011. © Springer-Verlag Berlin Heidelberg 2011

Optimal Power Flow for Indian 75 Bus System Using Differential Evolution

2

111

Optimal Power Flow (OPF) Problem Formulation

The OPF problem is to optimize the steady state performance of the power system in terms of the objective function while satisfying several equality and inequality constraints. OPF is a highly non-linear, non-convex, large scale static optimization problem due to large number of variables and limit constraints. It minimizes a certain objective function subjected to a set of equality and inequality constraints. Mathematically, the problem was defined as Minimize f(x, u, p) Subjected to g(x, u, p) = 0 h(x, u) ≤ 0 Where

x T =[δ,V]

u T =[VG1 ...VGNG ,PG2 ...PGNG ,T1...TNT ,QC1 ...QCNC ] The minimization function ‘f’ (objective function) can take different forms.

2.1

Objective Functions

Fuel Cost (FC) Minimization For active power optimization (economic load dispatch of the thermal units) fuel cost is considered as the objective function. The minimization function f can be obtained as sum of the fuel costs of all the generating units. NG

f = Fi (PGi )

(1)

i=1

Fi (PGi )=a i +bi PGi +ci PGi 2

(2)

Where ai, bi, ci are cost coefficients of unit-i, PGi is real power generation of unit-i. Real power losses (PL) For reactive power optimization, system transmission loss minimization is considered as the objective function. Transmission power loss in each branch is calculated from the load flow solution. The converged load flow solution gives the bus voltage magnitudes and phase angles. Using these, active power flow through the lines can be evaluated. Net system power loss is the sum of power loss in each line. Nl

f =  Loss i i=1

Where Nl is the number of transmission lines in a power system.

(3)

112

A.K. Das et al.

2.2

OPF Problem Constraints [8]

Equality constraints: These are load flow (nodal power balance) equations. Inequality Constraints: These constraints represent system Operating limits. Generator Constraints: Generator Voltage magnitudes VG, Generator active power PG and reactive power QG are restricted by their lower and upper limits. Transformer Constraints: Transformer taps have minimum and maximum setting limits. Switchable VAR sources: The switchable VAR sources have minimum and maximum restrictions on reactive power of Switchable VAR sources. Security constraints: These include the limits on the load bus voltage magnitudes and line flow limits.

3

Overview of Differential Evolution

Differential evolution algorithm, proposed by Storn and Price [9] in 1995, is a simple but effective population-based stochastic search technique for solving global optimization problems. The algorithm is named differential evolution (DE) owing to a special kind of differential operator, which they invoked to create new off-spring from parent chromosomes instead of classical crossover or mutation. The main differences between Genetic Algorithm and DE algorithm are the selection process and the mutation scheme that makes DE self-adaptive.

S ⊂ ℜn be the D dimensional search space of the problem under construc tion. The D.E. evolves a population of NP, D-dimensional individual vectors, X i ,G = Let

(xi,1 , xi, 2 ,  , xi,D ) ∈S , i = 1,2  , NP

from one generation to another. The primary

features of D.E can be stated as follows [9, 10]: 3.1

Initialization st

The 1 step of D.E. Algorithm is the initialization of the population. The population





should ideally cover the entire search space upper and lower bounds X max and X min :

  X max = {x1,max , x2,max ,......, xD,max } and X min = {x1,min , x2,min ,.......xD,min } .

th

The

th

j component of the i individual is initialized as follows:

xi, j = x j ,min + rand i j (0,1) ( x j ,max − x j ,min ) ; j ∈ [1,D] j

Here rand i (0,1) is a uniformly distributed random number within [0, 1].

(4)

Optimal Power Flow for Indian 75 Bus System Using Differential Evolution

3.2

113

Mutation



For every target vector, X i ,G , in any generation G, a mutant vector

 Vi,G is generat-

ed. The most common mutation policies generally used in DE are:









1. DE/rand/1/bin: V i ,G = X r1,G + F .( X r1,G − X r 3,G )



2. DE/best/1/bin: Vi,G

(5)

   = X best,G + F ⋅ ( X r1,G − X r 2,G ) 

3. DE/target-to-best/1/bin: Vi,G

(6)

     = X i,G + F.(X best,G − X i,G ) + F1.( X r1,G − X r 2,G )

(7)

Where r1, r2, and r3 are random and mutually exclusive integers generated in the range [1, NP], which should also be different from the trial vector’s current index i. F  and F1 are weight factors in [0, 2] for scaling differential vectors and X best ,G is the individual vector with best fitness value in the population at generation G. 3.3

Crossover

This operation involves binary crossover between the target vector mutant vector

 Vi ,G produced

 X i ,G and

the

in the previous step which produces the trial vector

 U i ,G = (ui ,1, G , ui , 2, G ,  , ui , n , G ) . The crossover operation is done as follows. v ui, j,G =  i,j,G  xi,j,G

if rand( 0 ,1 ) ≤ CR or j = jrand

(8)

otherwise

where CR is a user-specified crossover constant in the range [0, 1) and jrand is a randomly chosen integer in the range [1, NP] to ensure that the trial vector differ from its corresponding target vector

3.4

 X i ,G

 U i ,G will

by at least one parameter.

Selection



The fitness value of each trial vector f ( U i ,G ) is compared to that of its corresponding



target vector f ( X i ,G ) in the current population and the population for the G+1 generation is formed as follows: (for a minimization problem)

114

A.K. Das et al.

 Ui,G  X i ,G+1 =   X i,G

if

  f (Ui,G ) ≤ f ( X i,G )

otherwise

(9)

Where f (.) is the objective function.

4

Hybridized D.E and I.W.O with Population Exchange

The Invasive Weed Optimization (IWO) and DE algorithms are applied simultaneously on two sub-populations and population exchange is incorporated to refine the quality of the population after every generation. Invasive Weed Optimization [11] is a population-based algorithm based on trial and error method that copies the colonizing behavior of weeds [12].The main concept behind this population exchange is taken from the basic concept of any sports team, that, if the less-performing members of a team are replaced by the better-performing members, the team performs much better on an average. Thus the concept of population exchange rises. Suppose the 2 sub-populations are updated every generation according to IWO and DE respectively. Now, after every generation(G), the best (i.e. with best fitness) “m” percent members of the 1st sub-population, after that iteration(G), is exchanged with the worst(i.e. with worst fitness) “m” percent members of the 2nd sub-population, thereby refining the 2nd population and making it more effective for the next generation(G+1). The same exchange can be done to the 1st sub-population to make the algorithm perform in a better way as a whole. The value of “m” is taken as an input and made equal to 25. •

5

1st/ 2nd Sub-population for (G+1)th generation= m% of the best members of the 2nd/1st sub-population which replaces the worst m% + (100-m)% of the (10) remaining members from the (G)th generation.

Parameter Values Used for the Experiment

The parameters for in this particular experiment performed are: 1. 2.

Population Size: The total size of the population (NP) is taken to be 50. Weight Factor : F and F1 are both taken as F= F1= F_weight + rand (0, 1)*F_weight; F_weight =0.5

3. 4.

(11)

Crossover Constant : The value of the Crossover constant (CR) is taken as 0.7. Mutation Scheme of DE: The mutation scheme in the DE part of the algorithm applied to the superior sub-population is the “DE/target-to-best/1/bin” scheme that follows the mutation according to equation (7).

Optimal Power Flow for Indian 75 Bus System Using Differential Evolution

5. 6.

6

115

Limiting Values of Standard Deviation for IWO: The values of sdmin and sdmax are 0.001 and 5 respectively. The Modulation Index for IWO: The value of modulation index(pow) is taken as 4.5 (a sufficiently high value so as to magnify the search space for formation of new plants in IWO).

Results of the Algorithm Implementation

The Indian 75 Bus, 97 branch system consists of fifteen generators, twenty-four transformers, ninety-seven lines and twelve shunt reactors. In the optimization of OPF of the system there are sixty four (64) control variables of the system: 15 generator bus voltage magnitudes, 14 generator active powers (excluding the slack generator), 24 tap transformer settings and 12 shunt reactors. . The results given are the best solutions (minimum cost value) over 10 runs. Table 3 shows the control variables values and Table 4 shows the comparative fuel cost values between Hybridized D.EI.WO, D.E, P.S.O and I.W.O. Fig. 1 shows the convergence characteristics of the Differential Evolution (DE) algorithm. Table 1. Generator Cost Coefficients for Indian 75 bus system

Generator

Real Power Output Limit (MW)

No.

Maximum

Minimum

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

15 3 2 1.7 2.4 1.2 1 1 5.7 2.5 2.0 18 9 1.5 4.54

1 1 0.4 0.4 0.02 0.01 0.01 0.2 0.6 0.3 0.4 0.8 0.5 0.1 0.2

Fuel Cost Coefficients a 0.008 0.0014 0.0016 0.0016 0.0016 0.0018 0.0018 0.0018 0.0012 0.0017 0.0016 0.0004 0.0007 0.0015 0.0010

b 0.814 1.3804 1.5662 1.6069 1.5662 1.7442 1.7755 1.7422 1.1792 1.6947 1.6208 1.4091 0.6770 1.4910 1.0025

c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

116

A.K. Das et al. Table 2. Limits of Variables for 75 Bus System Sl .No.

Description

Upper Limit

Lower Limit

1.

Generator Voltage

1.1

0.95

2.

Load Voltage

1.05

0.9

3.

Transformer Tap

0.9

1.1

4.

Shunt Reactors

0.0

0.05

Table 3. Control Variable Values corresponding to the best result

Generator Voltages

Generator Powers

Tap Transformers

Shunt Reactors

Generator No.

Value (Volt)

Generator No.

Real Power (MW)

Reactive Power (MVAR)

Tap No.

Value

Shunt No.

Value

1

1.0300

2

100.000

89.400

0.982

0.9759

3

148.288

235.482

2

1.078

3

1.0139

4

71.0260

119.312

3

0.955

4

1.0536

5

43.8954

73.618

5

1.0627

6

7.07132

11.155

4

1.072

6

1.0696

7

26.2905

47.048

5

0.960

7

0.9874

8

25.9262

47.242

6

0.950

8

0.9516

9

61.7397

114.424

7

1.008

9

1.0937

10

60.0000

75.072

8

1.046

10

1.0691

11

40.1591

70.799

9

1.100

11

1.0973

12

40.0000

67.392

10

0.984

12

0.9919

13

113.202

51.437

13

1.0213

14

50.0000

35.600

11

1.070

14

1.0703

15

24.2167

36.987

12

1.080

15

0.9500

0.9000 1.0250 1.0875 0.9000 1.0875 1.0375 1.0625 1.0375 1.0750 1.0000 1.0000 1.0125 1.1000 0.9125 1.0250 0.9375 1.0500 1.0250 1.0875 1.0375 1.0625 1.0250 1.1000 1.0500

1

2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

The Losses of the system which was calculated by the 1st generator (slack generator) is found to be equal to 6.182217 MW.

Optimal Power Flow for Indian 75 Bus System Using Differential Evolution

117

Table 4. Comparison of the Best, Worst, Average and S.D of cost over 10 runs (Between hybridized D.E-I.W.O, D.E, P.S.O and I.W.O) Cost($/hr) Best Average Worst Standard Devn.

Hybrid D.E.-I.W.O 1042.97812 1071.22352 1117.89166 31.4594

Algorithm D.E. P.S.O 1087.54028 1261.8568 1124.67659 1306.6439 1209.0989 1344.0017 34.39535 47.0876

I.W.O 1213.8650 1265.9812 1296.0948 42.8861

Fig. 1. Best Value of Cost vs No of Generations (best run)

7

Conclusions

This paper presents a DE solution to the optimal power flow problem and is applied to an Indian 75-bus power system. The main advantage of DE over other modern heuristics is modeling flexibility, sure and fast convergence, less computational time than other heuristic methods. And it can be easily coded to work on parallel computers. The Future work in this area consists of applying other variants of D.E to the bus system and comparing the values obtained to the values obtained in this experiment.

References 1. Wood Allen, J., Bruce, F.: Power Generation Operation and Control. John Wiley and Sons, Inc., New York (1984) 2. Dommel, H.W., Tinney, W.F.: Optimal power flow solutions. IEEE Trans. Power App. Syst. 87(10), 1866–1876 (1968) 3. Sbalzarini, I.F., Muller, S., Koumoutsakos, P.: Multiobjective optimization using evolutionary algorithms, Center for Turbulence Research, Proceedings of the summer program (2000) 4. Osman, M.S., Abo-Sinna, M.A., Mousa, A.A.: A solution to the optimal power flow using genetic algorithm. Appl. Math. Comput. (2003)

118

A.K. Das et al.

5. Bakirtzis, A.G., Biskas, P.N., Zoumas, C.E., Petridis, V.: Optimal Power Flow by Enhanced Genetic Algorithm. IEEE Trans. on Power Syst. 17(2), 229–236 (2002) 6. Lai, L.L., Ma, J.T., Yokoyama, R., Zhao, M.: Improved genetic algorithms for optimal power flow under both normal and contingent operation states. Electrical Power & Energy Systems 19(5), 287–292 (1997) 7. Abido, M.A.: Optimal Power Flow using Particle Swarm Optimization. Electrical Power and Energy Systems 24(7), 563–571 (2002) 8. Sailaja Kumari, M., Sydulu, M.: An Improved Evolutionary Computation Technique for Optimal Power Flow Solution. International Journal of Innovations in Energy Systems and Power 3(1), 32–45 (2008) 9. Storn, R., Price, K.: Differential Evolution–A Simple and Efficient Heuristic for Global Optimization over Continuous Spaces. Journal of Global Optimization 11(4), 341–359 (1997) 10. Das, S., Suganthan, P.N.: Differential Evolution: A Survey of the State-of-the-art. IEEE Trans. on Evolutionary Computation 15(1), 4–31 (2011) 11. Mehrabian, A.R., Lucas, C.: A novel numerical optimization algorithm inspired from weed Colonization. Ecological Informatics, 355–366 (2006) 12. Dekker, J.: Course Works of Agronomy 517: A course on the biology and evolutionary ecology of invasive weeds (2005)

A Modified Differential Evolution Algorithm Applied to Challenging Benchmark Problems of Dynamic Optimization Ankush Mandal, Aveek Kumar Das, and Prithwijit Mukherjee Jadavpur University, ETCE department, Kolkata-700032, India {ankushmandal19,aveek23}@gmail.com, [email protected]

Abstract. Many real-world optimization problems are dynamic in nature. In order to deal with these Dynamic Optimization Problems (DOPs), an optimization algorithm must be able to continuously locate the optima in the constantly changing environment. In this paper, we propose a multi-population based differential evolution (DE) algorithm to address DOPs. This algorithm, denoted by pDEBQ, uses Brownian & adaptive Quantum individuals in addition to DE individuals to increase the diversity & exploration ability. A neighborhood based new mutation strategy is incorporated to control the perturbation & there by to prevent the algorithm from converging too quickly. Furthermore, an exclusion rule is used to spread the subpopulations over a larger portion of the search space as this enhances the optima tracking ability of the algorithm. Performance of pDEBQ algorithm has been evaluated over a suite of benchmarks used in Competition on Evolutionary Computation in Dynamic and Uncertain Environments, CEC’09. Keywords: Differential Evolution, Dynamic Optimization, Neighborhood based Mutation Strategy.

1 Introduction The Differential Evolution (DE) [1-3] has become one of the most powerful tools in evolutionary algorithms (EAs) in recent years. However, unlike the traditional EAs, the DE-variants perturb the current-generation population members with the scaled differences of randomly selected and distinct population members. Therefore, no separate probability distribution has to be used, which makes the scheme selforganizing in this respect. Several problems that we face in real world are dynamic in nature. For these Dynamic Optimization problems (DOPs), the function landscape changes over time. Practical examples of such situations are price fluctuations, financial variations etc. However, for dynamic environment, convergence problem is a significant limitation in case of EAs; if it becomes greatly converged, it will be unable to respond effectively to dynamic changes. So, in case of DOPs our main challenge is to maintain the diversity and at the same time to produce high quality solutions by tracking the moving optima. In this paper, a multi-population based partial DE variant with Brownian & Quantum individuals, denoted by pDEBQ, is proposed to address DOPs. The main change that we B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 119–126, 2011. © Springer-Verlag Berlin Heidelberg 2011

120

A. Mandal, A.K. Das, and P. Mukherjee

introduce in pDEBQ algorithm is the partial DE method. In each subpopulation, one individual is adaptive Quantum individual, analogous to particles in Quantum mechanics & one individual is Brownian individual whose trajectory is similar to the trajectory of any particle in Brownian motion; other individuals in the subpopulation evolve according to DE with a neighborhood based double mutation strategy. The double mutation strategy is developed to prevent the algorithm from converging quickly. In order to produce high accuracy in locating optima within the continuous search space, we have also introduced a control parameter to control the diversity. However, in case of EAs, stagnation is another significant problem as it reduces adaptability & exploration ability. So, we incorporated an aging mechanism to prevent the algorithm from stagnation. Also an exclusion scheme is used so that the subpopulations become evenly distributed over the entire search space. This increases the exploration efficiency & optima tracking ability of the algorithm. We tested the performance of our algorithm on a suite of DOPs generated by the Generalized Dynamic Benchmark Generator (GDBG) [4] which was used in Special Session & Competition on “Evolutionary Computation in Dynamic and Uncertain Environments”, CEC 2009.

2 Overview of Differential Evolution Differential Evolution was first proposed by Storn & Price [1]. The main steps of DE algorithm are: 1) Initialization of the population – the search process starts with a initial population of target vectors, 2) Mutation – a mutant vector is created corresponding to each target vector following some mutation scheme, 3) Crossover operation - in this operation the mutant vector exchanges its component with the target vector to form a trial vector, 4) Selection operation – the trial vector is selected for the next generation if it shows a better fitness value than the target vector. These steps (except the Initialization step) are repeated generation after generation until a stopping criterion is fulfilled. A detailed survey on DE is given in [3].

3 The pDEBQ Algorithm 3.1 Partial DE Scheme In order to ensure the population diversity, we introduced adaptive Quantum and Brownian individuals in addition to DE individuals. These Quantum and Brownian individuals do not follow the same rule as DE individuals to generate the next generation individuals. Hence, the population diversity is maintained to a greater extent. Actually, within a subpopulation, two individuals are randomly chosen at each generation; we apply Quantum individual generation process to one of them and Brownian individual generation process to the other. If one of the chosen individuals is the best individual in the subpopulation then we randomly choose another individual and replace it with the best individual. 1) Adaptive Quantum Individuals: In quantum mechanics, due to the uncertainty in position measurement, position of a particle is probabilistically defined. Here we use this idea to generate individuals within a specific region around the local best position.

A Modified Differential Evolution Algorithm

121

The steps for stochastically generating an individual whose position is inside a  sphere of radius R and centered on the local best position Lb are as follows: i) Generate a radius randomly from a uniform distribution within the range (0, R). r = U (0, R) ii) Generate a vector whose each component is randomly taken from a normal  distribution having mean=0 and variance=1. X = {x1 , x2 ,......., x D } ; x d ∈ N (0, 1); where 1 ≤ d ≤ D iii) Compute the distance of the vector from the origin. Dist = iv) New quantum individual’s position will be:   Lb + (r / Dist) * X



D 2 x i =1 i

(1)

The probability of generating a point near the local best point increases with dimensionality. In pDEBQ, the quantum individuals are adaptive because the radius within which the individuals are to be generated is self adaptive in nature, i.e. the radius is varied automatically according to the condition. 2) Brownian Individuals: Brownian motion is the mathematical model used to describe the random movement of particles suspended in a fluid. We used a very simple model to incorporate the Brownian motion into the positions of individuals. Actually, new individuals are generated within a Gaussian hyper parallelepiped. If the  local best position is Lb then the new Brownian individual’s position will be:   Lb + N (0, σ ) (2) where σ is the standard deviation of the normal distribution from which the perturbation is randomly sampled. 3) DE Individuals: These individuals follow the DE method for generating new individuals. Here we have developed a new mutation scheme to produce mutant vectors. We have discussed this mutation scheme in the following section in details. 3.2 Double Mutation Strategy In the case of DOPs, the optima positions are changed dynamically. If the population becomes greatly converged to a global optima then it will lose its ability to find the global optima again when the position of the global optima changes. So, here the idea is to control the perturbation to slow down the searching process. For this, we have developed a double mutation scheme. In this scheme, we first generate a mutant vector according to a neighborhood based mutation scheme and then we add this mutant vector to the local best vector with a weight factor having maximum value of 0.1. As we did the mutation process twice for each DE individuals, we call it double mutation strategy. 1) Neighborhood based Mutation Scheme: It’s a modified DE/current-to-best/2/bin scheme. In order to overcome the limitations of fast but less reliable convergence performance of DE/current-to-best/2/bin scheme, we have made some changes in the process of generating two differential vectors. For the 1st differential vector, in our

122

A. Mandal, A.K. Das, and P. Mukherjee

modified scheme we used the difference between nearest memory individual and the current individual where the memory archive is the collection of best individuals from the previous subpopulations. This change is done to control the convergence of the population towards global optima and encourage the subpopulations to explore the vicinity of local best position. For the second differential vector, we took the difference between the best individual in the neighborhood and the worst individual in the neighborhood with respect to the current individual. This second modification is done to guide the mutant vector to explore the neighborhood of the current individual within the subpopulation. During the process of generating the mutant vector, the scaling factor is independently generated for each dimension of each differential vector and its value also depends on the magnitude of the differential vector along the corresponding dimension. Actually, for each dimension of each differential vector, the scaling factor is randomly generated from a uniform distribution within a range and this range is varied inversely with the magnitude of the differential vector along the corresponding dimension. This adaptation of the scaling factor is done to amplify the differential vector properly. The first mutation according to neighborhood based mutation scheme can be expressed as: j j j j j j j j Vmut ,G = X i,G + Fm ⋅ ( X m,G − X i,G ) + Fbw ⋅ ( X best,G − X worst,G ) ,where j ∈ {1,2,......., D} (3)   Here, X i ,G is the current vector, X best ,G is the best vector in the neighborhood with respect to the current vector. It’s the vector within the corresponding subpopulation  1  f ( X k ,G )  −1 (k = 1, 2,…, M where M = Number of individuals in the for which  rik  f ( X i ,G )  subpopulation and k ≠ i) is maximum. Here, rik is the distance between the vectors     X i ,G and X k ,G . It is calculated as X k ,G − X i ,G .

 Similarly X worst ,G is the worst vector in the neighborhood with respect to the  f ( X k ,G )  1  (k = 1, 2,…, M where M = Number current vector. For this vector, 1−  rik  f ( X i ,G )   of individuals in the subpopulation and k ≠ i) is maximum among all individuals within the subpopulation. Here also rik has the same meaning as previously.   X m,G is the nearest memory individual to X i ,G in terms of Euclidean distance. The scaling factors are generated as follows:  | x mj ,G − x ij,G Fmj = 0.3 + 0.7 ∗ rand ( 0,1) ∗  1 −  | x Rj | 

|   

j j  | x best ,G − x worst ,G Fbwj = 0.3 + 0.7 ∗ rand (0,1) ∗ 1 −  | x Rj | 

where | x Rj | is the search range.

|   

(4)

(5)

A Modified Differential Evolution Algorithm

123

2) Second Stage Mutation: As mentioned earlier, we add the mutant vector from the 1st stage of mutation to the local best vector with a weight factor. In this way we perturb the local best vector in a controlled manner. The second mutation can be expressed as:    (6) V final ,G = (1 − ω ) ⋅ Lb,G + ω ⋅ V mut ,G

 Where Lb,G is the local best vector, i.e. the best vector of the corresponding  subpopulation, Vmut ,G is the mutant vector generated from 1st stage mutation and ω is the weight factor. 3.3 Exclusion Rule

For optimizing DOPs, it is important to ensure that the population is almost evenly distributed over the entire search space. If it is not so then the population will lose its exploration capability. In order to achieve the uniform distribution in this multipopulation algorithm, we developed an exclusion rule to ensure that different subpopulations are located around different basins of attraction. Here, the strategy is to calculate the Euclidean distance between each pair of best individuals of two different subpopulations at each generation. If the distance between the best individuals of any two subpopulations falls below the marginal value then the subpopulation having the best individual of lower fitness value among the two is marked for re-initialization. The marginal value of the distance is calculated according to the following rule: If the search range is R, the search space is D dimensional and there are SP number of subpopulations then the marginal value for the distance is Dis_marginal ≤ R /( SP * D ). 3.4 Ageing Mechanism

It may happen that a subpopulation gets trapped in some local optima. This stagnation phenomenon hinders the search process. So, in order to get rid of stagnation at any local optima, we employed a simple aging strategy. Consistent bad performance of any individual is also taken into account in this strategy. Algorithm for Aging Mechanism:Considering i-th individual of j-th subpopulation. 1. if j-th subpopulation have the global best individual then do not re-initialize the subpopulation. 2. else if i-th individual is the best individual in j-th subpopulation then Age_best(j,i)=Age_best(j,i)+1. if Age_best(j,i)≥30 then re-initialize j-th subpopulation and reset Age_best(j,i) to 0. 3. else if i-th individual is the worst individual in j-th subpopulation then Age_worst(j,i)=Age_worst(j,i)+1. if Age_worst(j,i)≥20 then re-initialize the individual and reset Age_worst(j,i) to 0. (Initially all the entries of Age_best & Age_worst were 0)

124

A. Mandal, A.K. Das, and P. Mukherjee

3.5 Control Parameter

In order to locate the global optima within the current function landscape, we introduced a control parameter. Depending on the condition this parameter can have a value 0, 1 or 2. We measured the progress of the algorithm every 20 iterations. The control parameter (C) is set to the value corresponding to the predefined range in which the progress falls. If C becomes 1 then the Quantum individuals are not generated; if C becomes 2 then the Brownian individuals are not generated and if C becomes 0 then both Quantum and Brownian individuals are generated. Actually, this parameter controls the diversity of the population in order to give high accuracy in locating global optima.

4 Experimental Settings CEC 2009 benchmark problems for dynamic optimization use the generalized dynamic benchmark generator (GDBG) proposed in [5], which constructs dynamic environments for the location, height and width of peaks. In these benchmark problems, the basic test functions are as follows: 1) F1: Rotation peak function, 2) F2: Composition of Sphere’s function, 3) F3: Composition of Rastrigin’s function, 4) F4: Composition of Griewank’s function, 5) F5: Composition of Ackley’s function, 6) F6: Hybrid Composition function. Only F1 is maximization problem and others are minimization problems. In F1, there are two tests, one using 10 peaks and another using 50 peaks. There are seven change types for each test functions. However, we tested our algorithm over two most difficult change types for each test function which are large step change and random change. In these change types other algorithms gave the worst results. A detailed description about the experimental settings is given in [4].

5 Parameter Settings 1. Population Size: The Population size (NP) was 60. We divided the whole population into 10 subpopulations. Dimensionality was 10 for all the test functions. 2. Control Parameter Adaptation: First we define the difference of the global best fitness values between first 20 iterations as PR. From this point onwards, if the global best fitness values between 20 iterations have a difference greater than PR then the current value of PR is replaced by this new value. If the difference becomes less than (PR /10) but greater than (PR /50) then C is set to 1. If the difference becomes less than (PR /50) then C is set to 2. 3. Quantum Individual Adaptation: The parameter R which is the radius within which the quantum individuals are generated is adaptive in nature. If C is 0 then R is set to 1. If C is 2 then R changes according to the following rule depending upon the difference (Diff) of global best fitness values between 20 iterations.

 PR R = Diff ∗ log10 10 + 50 * Diff 

  

(7)

A Modified Differential Evolution Algorithm

125

4. Marginal Value for Distance between Best Individuals: If C is 0 then Dis_marginal is set to 0.08 but if C becomes 1 or 2 then Dis_marginal becomes 0.03. 5. Crossover Probability: The crossover probability (CR) was kept fixed at 0.9 throughout the search process. 6. Weight Factor for Double Mutation Strategy: The weight factor ω was set to 0.035 for maximization problem (i.e. F1) and 0.1 for minimization problems.

6 Experimental Results In this section, we represent the performance of our algorithm in terms of mean and standard deviation (STD) of error values obtained for 14 significantly difficult test Table 1. Comparison of Error Values for Change Type T2 Function F1 (10 Peaks) F1 (50 Peaks) F2 F3 F4 F5 F6

Error Value Mean STD Mean STD Mean STD Mean STD Mean STD Mean STD Mean STD

pDEBQ 1.6451e-007 2.7157e-007 5.0230e-008 7.3943e-008 4.4666 7.5912 750.1274 193.5187 18.7268 25.2722 0.16270 1.4162 9.5653 27.3228

Algorithm DASA jDE 4.18 3.5874 9.07 7.83849 4.86 4.08618 7.0 6.4546 25.6 43.004 83.2 114.944 824 558.497 204 384.621 65.6 49.5044 160 135.248 0.99 0.33392 4.05 1.64364 37 10.3083 122 12.2307

CESO 2.36549 7.93215 4.35869 6.21243 12.1431 47.2519 791.1642 254.1579 28.5925 98.6229 2.282617 4.4479 20.0895 60.9701

Table 2. Comparison of Error Values for Change Type T3 Function F1 (10 Peaks) F1 (50 Peaks) F2 F3 F4 F5 F6

Error Value Mean STD Mean STD Mean STD Mean STD Mean STD Mean STD Mean STD

pDEBQ 6.2113e-009 9.0601e-009 0.7390 1.7022 3.7863 6.2560 750.0162 230.8435 13.2794 23.7620 0.29820 1.7355 10.25390 30.1322

Algorithm DASA jDE 6.37 2.99962 10.7 7.12954 8.42 4.29209 9.56 6.74538 18.9 50.1906 67.8 124.015 688 572.105 298 386.09 53.6 51.9448 140 141.78 0.949 0.3579 3.31 1.83299 26.7 10.954 98.4 23.2974

CESO 5.17873 8.97652 6.26213 9.16489 11.5219 43.6323 634.5213 341.2314 23.4561 92.5379 2.81789 5.25534 18.5418 65.6901

126

A. Mandal, A.K. Das, and P. Mukherjee

cases. We have also compared these results with three significant evolutionary DOP solvers - CESO [6], jDE [2], and DASA [7]. We represented it in Table 1& 2.

7 Conclusion In this paper, a partial differential evolution algorithm namely pDEBQ has been proposed to address DOPs. Key features of the algorithm are briefly outlined as follows:      

It uses a partial DE scheme which obviously shares the traditional DE framework. In addition to DE individuals, it uses adaptive Quantum and Brownian individuals to increase the diversity and exploration ability. A control parameter is introduced to control the diversity as necessary. The algorithm also uses an aging mechanism to get rid of stagnation. The DE individuals produce the donor vectors according to a neighborhood based double mutation strategy to control the perturbation. An exclusion scheme is used so that the subpopulations become evenly distributed over the entire search space.

The statistical summary of the simulation results also shows that pDEBQ is far better than other algorithms in terms of performance in dynamic landscape. So, we can easily conclude that pDEBQ is very good optimizer for DOPs. Our future plan includes more co-operation and information exchange between the subpopulations. It will also be valuable to try to make the crossover probability adaptive to the condition of the fitness landscape.

References 1. Storn, R., Price, K.: Differential evolution: A simple and efficient heuristic for global optimizationover continuous spaces. Journal of Global Optimization 11, 341–359 (1997) 2. Brest, J., Zamuda, A., Boskovic, B., Maucec, M.S., Zumer, V.: Dynamic Optimization using Self-Adaptive Differential Evolution. In: Proc. 2009 Cong. on Evol. Comput., pp. 415–422 (2009) 3. Das, S., Suganthan, P.N.: Differential evolution – a survey of the state-of-the- art. IEEE Transactions on Evolutionary Computation 15, 4–31 (2011) 4. Li, C., Yang, S., Nguyen, T.T., Yu, E.L., Yao, X., Jin, Y., Beyer, H.-G., Suganthan, P.N.: Benchmark Generator for CEC 2009 Competition on Dynamic Optimization, Technical Report, University of Leicester, University of Birmingham, Nanyang Technological University (September 2008) 5. Li, C., Yang, S.: A Generalised Approach to Construct Benchmark Problems for Dynamic Optimization. In: Li, X., Kirley, M., Zhang, M., Green, D., Ciesielski, V., Abbass, H.A., Michalewicz, Z., Hendtlass, T., Deb, K., Tan, K.C., Branke, J., Shi, Y. (eds.) SEAL 2008. LNCS, vol. 5361, pp. 391–400. Springer, Heidelberg (2008) 6. Lung, R.I., Dumitrescu, D.: A collaborative model for tracking optima in dynamic environments. In: Proc. 2007 Congr. Evol. Comput., pp. 564–567 (2007) 7. Korosec, P., Silc, J.: The differential anti-stigmergy algorithm applied to dynamic optimization problems. In: Proc. 2009 Congr. Evol. Comput., pp. 407–414 (2009)

PSO Based Memetic Algorithm for Unimodal and Multimodal Function Optimization Swapna Devi, Devidas G. Jadhav, and Shyam S. Pattnaik National Institute of Technical Teachers’ Training & Research (NITTTR), Sector-26, Chandigarh, 160019 India [email protected], [email protected]

Abstract. Memetic Algorithm is a metaheuristic search method. It is based on both the natural evolution and individual learning by transmitting unit of information among them. In the present paper, Genetic Algorithm due to its good exploration capability is used for exploration and Particle Swarm Optimization (PSO) does local search. The memetic process is realized using the fitness information from the individual having best fitness value and searching around it locally with PSO. The proposed algorithm (PSO based memetic algorithm -pMA) is tested on 13 standard benchmark functions having unimodal and multimodal property. When results are compared, the proposed memetic algorithm shows better performance than GA and PSO. The performance of the discussed memetic algorithm is better in terms of convergence speed and quality of solutions.

1 Introduction Hybridization of Genetic Algorithm with a local search mechanism is known as Memetic Algorithms (MAs) [1]-[3]. It is inspired by both Darwinian principle of natural evolution and Dawkins’ notion of a meme as a unit of information or idea transmission from one to another or cultural evolution capable of individual learning [1][2]. In general, MA is a synergy of evolution and individual learning [1] [3], which is improving the capability of evolutionary algorithms (like GA) in finding optimal solutions accurately in function optimization problems with faster speed of convergence. Stochastic optimizers that have drawn the attentions of researchers in recent times are Genetic Algorithm, Differential Evolution, Evolutionary Programming etc. [4]-[6] where a population of the solutions is utilized in the search process. Many population based algorithms are capable of exploring and exploiting the promising regions in the search space but they take relatively longer time [1]. Hence, algorithms are combined for better exploration and exploitation making it faster as well as accurate. Memetic Algorithm is an outcome of such combination of GA with Local Search. The RealCoded Memetic Algorithm based on crossover-hill climbing maintains a pair of parents and performs crossover until offspring are obtained for the best offspring selection [3]. Some of the population based search algorithms like Particle Swarm Optimization (PSO) have a tendency of premature convergence [7]. To overcome this drawback, combinatorial optimization method based on particle swarm optimization and stochastic local search is designed in which stochastic search method takes the solution out from local trapping [7]. GA along with gradient-based information as a B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 127–134, 2011. © Springer-Verlag Berlin Heidelberg 2011

128

S. Devi, D.G. Jadhav, and S.S. Pattnaik

local search has been used to develop memetic algorithms [8]. But the gradient-based methods failed in the cases of multimodal and non-differentiable functions [8]. So, population based local search algorithms have advantages over the gradient type local searches for not getting trapped in local optima [9].PSO is population based and is good for local searching; therefore, it is used as a local search in the proposed algorithm. Genetic Algorithm is mainly used for exploration and the Particle Swarm Optimization (PSO) does the local search. After certain exploratory search by the genetic algorithm, the current best solution obtained is further refined by the local search. The proposed algorithm converges for multimodal functions also. The paper is organized as follows. In section 2 the proposed Memetic Algorithm (pMA) is presented. Section 3 gives the detail about experimentation and results and section 4 concludes the paper.

2 Memetic Algorithm with GA and PSO as Local Search The GA is used for an intelligent exploration having a random search confined within a defined search space for solving a problem optimally with the help of population [4]. Standard GA applies genetic operators such as selection, crossover, and mutation on initially generated random population in order to evolve a complete generation of new solutions [4] for exploration of the complete search space. GA is good for exploration but it is slow. So, local search is applied which searches for a local optima. This is called memetic algorithm and this enhanced algorithm shows speed of convergence. Elitism prevents the loss of good solutions once the solutions are found [4]. Elitism is used in algorithm for enhancing the performance. In proposed algorithm, GA is used mainly for exploration purpose and PSO is used for local search. The population is real coded for both genetic algorithm and local search algorithm [3][9]. The population is uniformly distributed random population in the specified range for respective functions as given in Table 1. Based on fitness, the current best individual is selected for elite preservation. Then the population goes through the GA operations viz., crossover and mutation. The crossovers used are BLX-α crossover [10] and SBX crossover [11]. The offspring O1= { o11, o12, . . . , o1i , . . . , o1d} and O2= { o21, o22, . . . , o2i , . . . , o2d} are generated from parents P1={p11, p12, . . . ,p1i, . . . , p1d} and P2={p21, p22, . . . ,p2i, . . . , p2d}having ‘d’ dimension. Table 1. Benchmark functions used for experimental study (D is the dimension of the functions) [6]

F F1 F2 F3 F4 F5 F6 F7

Function Sphere Schwefel’s 2.22 Schwefel’s 1.2 Schwefel’s 2.21 Rosenbrock Step Quartic

Range

NFFE

[-100,100]D [-10,10]D [-100,100]D [-100,100]D [-30,30]D [-100,100]D [-1.28,1.28]D

150000 200000 500000 500000 500000 150000 300000

F F8 F9 F10 F11 F12 F13

Function Schwefel’s 2.26 Rastrigin Ackley Griewank Penalized #1 Penalized #2

Range

NFFE

[-500,500]D [-5.12,5.12]D [-32,32]D [-600,600]D [-50,50]D [-50,50]D

300000 300000 150000 300000 150000 150000

PSO Based Memetic Algorithm for Unimodal and Multimodal Function Optimization

129

In BLX-α crossover O is chosen randomly between the interval [(P1 – I · α) , (P2 + I · α) ] with the condition P1 < P2 and I = max(P1 , P2) - min(P1 , P2 ). In SBX crossover the effect of the one-point crossover of the binary representation is tried to emulate. The crossover generates two offspring O1=1/2[(1+B) P1 + (1-B) P2] and O2=1/2[(1-B) P1 + (1+B) P2] where B ≥0 is a sample from random generator having density function shown in equation (1). 1 η (η + 1)B , if 0 ≤ B ≤ 1 2 1 1 (η + 1) , if B > 1 +2 η 2 B

p(B) =

(1)

This distribution is obtained by using uniform random number source u(0,1) as shown in equation (2). 1

B(u) = (2u ) η +1 ,

if u ≤

1 2

(2)

1

1 ( 2(11−u ) ) η +1 , if u > 2

where η=2 is used for better results. Exploitation capacity of operator increases with higher values of η. Mutation changes the variable randomly under uniform normal distribution in the specified range as per Table 1. The local search algorithm is evoked after a decided number of iterations or generations of genetic algorithm (known as Glocal) are elapsed. A small population is generated around the current best individual by perturbations and given to PSO local search. Velocity and position are updated by equation (3) and (4) respectively.

(

v id = w ∗ v id + c1 ∗ rand 1id pbest id − X id

(

+ c 2 id ∗ rand 2 id ∗ gbest d − xid

X id = X id + vid

)

)

(3) (4)

Initially when error is above 0.01 the local search intensity is less i.e., NFFE=200 and when it is below 0.01, intensity is increased by 50% of the previous one. The PSO ensures the particles to be within the range. The performance of the pMA-BLX-α, pMA-SBX, PSO and GA is checked with 13 standard benchmark functions shown in Table 1. The proposed algorithm is applied on unimodal and multimodal standard benchmark functions and the performance is noted by determining accuracy and convergence speed. Pseudo Code of MA-PSO is as follows: 1.Initialization: Generate a random initial population 2.while Stopping conditions are not satisfied do 3.Evaluate all individuals in the population 4.Find current best individual (1 elite individual) 5.If Glocal=2 for best individual do (refinement by PSO) 6.Generate local population by elite perturbation 7.While stopping conditions are not satisfied do 8.Evaluate the population

130

S. Devi, D.G. Jadhav, and S.S. Pattnaik

9.Update current_best_individuals, current_Global_best 10.Calculate velocity and Update the positions 11.Make the particles feasible as per range, if any 12.endwhile 13.Return best individual to main algorithm 14.end if 15.Select individuals for crossover with probability Pc 16.Crossover parents by BLX-• or SBX crossover operators 17.Make the offspring feasible 18.Mutate individual with mutation probability Pm 19.Replace the parents by offspring preserving the elite 20.end while

3 Experimentation In the genetic as well as in memetic algorithms, a population of 100 individuals of real-valued representation is used. The crossover operator used is the BLX-α crossover or SBX crossover with crossover probability set to 0.8. For BLX-α crossover, α = 0.3 and for SBX crossover, η =2 are used. The mutation probability is set to 0.05. The algorithm is using generational replacement of individuals preserving one elite. Two stopping criterions are used – the number of fitness function evaluations (NFFE as in Table 1) or an error value of 10-8. PSO is also executed for the same dimensions, population, stopping criterion etc. In PSO, inertia weight is 0.4 and constants considered are c1=c2=2. All functions are having 10 dimensional inputs. All algorithms are executed for 25 independent trials and the best errors in trails are used for averaging and calculating standard deviation (Std Dev) presented in Table 2. Table 2. Comparison of the experimental results GA, PSO and PSO based Memetic Algorithm (pMA) using PSO as Local Search F F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13

GA-BLX-α Mean (Std Dev) 5.44E+2 (3.34E+2) 5.86E+0 (4.94E+0) 9.40E+2 (6.04E+2) 6.86E+0 (1.57E+0) 1.26E+4(1.27E+4) 5.09E+2(3.36E+2) 1.013E+0 (4.12E-1) 2.33E+2 (1.49E+2) 2.39E+1 (5.09E+0) 5.00E+0 (7.83E-1) 4.03E+0 (1.89E+0) 3.92E+2 (1.45E+3) 1.14E+4 (3.66E+4)

GA-SBX Mean (Std Dev)

PSO Mean (Std Dev)

pMA-BLX-α Mean (Std Dev)

pMA-SBX Mean (Std Dev)

1.57E+3 (4.25E+2) 3.76E+1 (2.16E+1) 7.10E+3 (2.72E+3) 6.59E+0 (1.88E+0) 6.88E+5 (5.03E+5) 1.54E+3 (4.74E+2) 9.84E-1 (4.23E-1) 5.68E+2 (1.83E+2) 2.59E+1 (5.69E+0) 4.65E+0 (8.18E-1) 9.40E+0 (3.11E+0) 1.18E+6 (1.40E+6) 6.58E+6 (3.99E+6)

7.53E-9(1.53E-9) 6.46E-8 (1.99E-7) 9.30E-9(7.45E-10) 9.52E-9 (5.46E-10) 3.60E+3 (1.80E+4) 0.00E+0(0.00E+0) 4.48E-4 (2.32E-4) 5.67E+2 (1.96E+2) 2.75E+0(1.36E+0) 4.62E-2 (2.31E-1) 6.37E-2(2.98E-2) 2.07E-2 (4.23E-2) 8.79E-4 (3.04E-3)

6.10E-9 (2.27E-9) 8.80E-09 (1.02E-9) 8.78E-09 (1.13E-9) 9.30E-09(5.75E-10) 1.95E-1 (8.00E-1) 0.00E+0 (0.00E+0) 3.76E-1 (2.22E-1) 4.60E+2 (3.49E+2) 1.59E-1 (3.72E-1) 5.70E-08 (1.98E-8) 1.24E-01 (7.57E-2) 5.21E-9 (2.10E-9) 6.22E-9 (2.58E-9)

6.77E-9 (2.36E-9) 8.62E-09 (8.77E-10) 9.17E-9 (8.31E-10) 9.28E-09 (5.59E-10) 9.61E-1 (1.74E+0) 0.00E+0 (0.00E+0) 3.30E-1 (2.18E-1) 6.65E+2 (2.53E+2) 1.59E-1 (3.72E-1) 6.39E-08 (1.83E-8) 1.47E-1 (7.71E-2) 5.06E-9 (2.72E-9) 5.96E-9 (2.50E-9)

PSO Based Memetic Algorithm for Unimodal and Multimodal Function Optimization

131

Table 3 and Table 4 show the error values achieved by algorithms, pMA-BLX-α and pMA-SBX, respectively in 25 runs for functions F1-F13. 3.1 Discussion These algorithms are tested on standard benchmark functions having unimodal, multimodal properties with noise and discontinuity. Functions F1-F5 are with unimodal property. Function F6 is a step function, which is having one minimum and is a discontinuous function. Function F7 is a noisy quartic function comprising of random value in range [0, 1). Functions F8 – F13 are multimodal functions in which the number of local minima is increasing exponentially with the problem dimension [6]. For unimodal functions, the convergence rates are more interesting than the final results of optimization. In case of multimodal functions, the final results are important since they reflect algorithm’s ability of escaping from poor local optima and locating a good near-global optimum. The pMA is converging quickly in most of the functions like F1-F4, F6, F9-F10, and F12-F13 (both unimodal and multimodal) as shown in Table 3 and Table 4. In Fig. 1 the graphs show the number of fitness function evaluations on x-axis and error in the optimum value averaged over 25 runs on the y-axis. For comparison the convergence Table 3. Error values achieved by pMA-BLX-α in 25 runs for functions F1-F13 F F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13

1st (Best) 2.22E-09 6.26E-09 5.69E-09 7.95E-09 3.75E-04 0.00E+0 1.21E-01 1.27E-04 3.71E-09 2.79E-08 2.96E-02 1.38E-09 7.78E-10

7th 4.73E-09 8.08E-09 8.49E-09 8.95E-09 4.98E-04 0.00E+0 1.88E-01 2.17E+2 7.55E-09 4.19E-08 7.39E-02 3.60E-09 5.06E-09

13th (Median) 6.23E-09 9.22E-09 9.04E-09 9.48E-09 6.22E-04 0.00E+0 3.44E-01 4.34E+2 8.62E-09 5.15E-08 1.01E-01 4.76E-09 6.87E-09

19th 8.21E-09 9.51E-09 9.47E-09 9.73E-09 8.63E-04 0.00E+0 5.22E-01 6.51E+2 9.82E-09 6.90E-08 1.60E-01 6.45E-09 8.28E-09

25th(Worst) 9.77E-09 1.00E-08 9.93E-09 9.94E-09 3.99E+0 0.00E+0 9.26E-01 1.05E+3 9.95E-01 9.59E-08 3.74E-01 9.49E-09 9.84E-09

Mean 6.10E-09 8.80E-09 8.78E-09 9.30E-09 1.95E-01 0.00E+0 3.76E-01 4.60E+2 1.59E-01 5.70E-08 1.24E-01 5.21E-09 6.22E-09

Std Dev 2.27E-09 1.02E-09 1.13E-09 5.75E-10 8.00E-01 0.00E+0 2.22E-01 3.49E+2 3.72E-01 1.98E-08 7.57E-02 2.10E-09 2.58E-09

Table 4. Error values achieved by pMA-SBX in 25 runs for functions F1-F13 F F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13

1st (Best) 2.51E-09 6.93E-09 7.18E-09 8.15E-09 3.95E-04 0.00E+0 9.03E-02 1.27E-04 4.90E-09 3.65E-08 3.74E-02 1.62E-09 1.47E-09

7th 5.18E-09 8.03E-09 8.86E-09 8.93E-09 6.28E-04 0.00E+0 1.58E-01 5.35E+2 7.88E-09 5.03E-08 9.59E-02 2.71E-09 3.54E-09

13th (Median) 7.04E-09 8.65E-09 9.60E-09 9.39E-09 7.66E-04 0.00E+0 3.23E-01 6.91E+2 8.98E-09 6.40E-08 1.21E-01 4.70E-09 6.13E-09

19th 9.07E-09 9.25E-09 9.76E-09 9.80E-09 3.91E-02 0.00E+0 3.66E-01 8.49E+2 9.79E-09 7.52E-08 1.99E-01 6.96E-09 7.92E-09

25th(Worst) 9.95E-09 9.84E-09 9.98E-09 1.00E-08 4.02E+0 0.00E+0 1.00E+0 1.16E+3 9.95E-01 9.80E-08 3.17E-01 9.50E-09 9.37E-09

Mean 6.77E-09 8.62E-09 9.17E-09 9.28E-09 9.61E-01 0.00E+0 3.30E-01 6.65E+2 1.59E-01 6.39E-08 1.47E-01 5.06E-09 5.96E-09

Std Dev 2.36E-09 8.77E-10 8.31E-10 5.59E-10 1.74E+0 0.00E+0 2.18E-01 2.53E+2 3.72E-01 1.83E-08 7.71E-02 2.72E-09 2.50E-09

132

S. Devi, D.G. Jadhav, and S.S. Pattnaik

Fig. 1. Average fitness error versus NFFE for functions F2, F3, F4, F5, F9, F10, F11& F13

PSO Based Memetic Algorithm for Unimodal and Multimodal Function Optimization

133

characteristics of GA-BLX-α, GA-SBX, pMA-BLX-α, pMA-SBX and PSO are plotted. As seen from the figure, the GA-BLX-α and GA-SBX are performing in a similar way and pMA-BLX-α and pMA-SBX are also having similar nature of convergence. PSO is converging better than GA but showing lower performance than pMA. The future work could be to integrate local search with improved PSO variants [12] and incorporating CEC 2005 test problems [13] for validation.

4 Conclusion In this paper a new variant of memetic algorithm named pMA is developed. Particle Swarm Optimization (PSO) is used as the local search in this Memetic Algorithm. The mechanism of PSO for dragging all the particles towards the best one shows the instating property of ‘meme’. PSO is good at local search and showing its usefulness in the pMA. The proposed PSO based Memetic Algorithm (pMA) is performing robustly as compared to Genetic Algorithms in terms of higher accuracy and convergence speed. PSO is also showing promising performance individually and supporting GA very well to make the MA. Both the algorithms are having convergence accuracy upto the error value 10-8 for most of the functions and showing better speed, success and consistency of the convergence. The variants of PSO can be integrated as a local search for making MA and can be tested on CEC2005 test problems.

References 1. Nguyen, Q.H., Ong, Y.S., Krasnogor, N.: A Study on the Design Issues of Me-metic Algorithm. In: Proc. of the IEEE Congr. Evol. Comput. (CEC 2007), pp. 2390–2397 (September 2007) 2. Moscato, P.A.: On evolution, search, optimization, genetic algorithms and martial arts: Towards memetic algorithms, Tech. Rep. Caltech Concurrent Computation Program, California Institute of Technology, Pasadena, CA, Report 826 (1989) 3. Lozano, M., Herrera, F., Krasnogor, N., Molina, D.: Real-Coded Memetic Algo-rithms with Crossover Hill-Climbing. Evolutionary Computation 12(3), 273–302 (2004) 4. Michalewicz, Z.: Genetic Algorithms + Data Structures = Evolution Programs. Springer, Heidelberg (1996) 5. Das, S., Suganthan, P.N.: Differential evolution - a survey of the state-of-the-art. IEEE Trans. on Evolutionary Computation 15(1), 4–31 (2011) 6. Yao, X., Liu, Y., Lin, G.: Evolutionary Programming Made Faster. IEEE Trans. on Evolutionary Computation 3(2), 82–102 (1999) 7. Akbari, R., Ziarati, K.: Combination of Particle Swarm Optimization and Stochastic Local Search for Multimodal Function Optimization. In: Proc. of the IEEE Pacific-Asia Workshop on Computational Intelligence and Industrial Application (PACIIA 2008), pp. 388– 392 (2008) 8. Li, B., Ong, Y.S., Le M.N., Goh, C.K.: Memetic Gradient Search. In: Proc. of the IEEE Congress on Evol. Comput. (CEC 2008), pp. 2894–2901 (2008) 9. Jadhav, D.G., Pattnaik, S.S., Devi, S., Lohokare, M.R., Bakwad, K.M.: Approximate Memetic Algorithm for Consistent Convergence. In: Proc. National Conf. on Computational Instrumentation (NCCI 2010), pp. 118–122 (March 2010)

134

S. Devi, D.G. Jadhav, and S.S. Pattnaik

10. Eshelman, L.J., Schaffer, J.D.: Real-coded genetic algorithms and interval-shemata. In: Darrell Whitley, L. (ed.) Foundation of Genetic Algorithms, vol. 2, pp. 187–202. Morgan Kaufmann, San Mateo (1993) 11. Deb, K., Agrawal, R.B.: Simulated binary crossover for continuous search space. Complex Syst. 9(2), 115–148 (1995) 12. Liang, J.J., Qin, A.K., Suganthan, P.N., Baskar, S.: Comprehensive Learning Particle Swarm Optimizer for Global Optimization of Multimodal Functions. IEEE Trans. on Evol. Comput. 10(3), 281–295 (2006) 13. Suganthan, P.N., Hansen, N., Liang, J.J., Deb, K., Chen, Y.P., Auger A., Tiwari, S.: Problem Definitions and Evaluation Criteria for the CEC 2005 Special Session on RealParameter Optimization. Technical Report, Nanyang Technological University, Singapore, & KanGAL Report #2005005, IIT Kanpur, India (May 2005)

Comparison of PSO Tuned Feedback Linearisation Controller (FBLC) and PI Controller for UPFC to Enhance Transient Stability M. Jagadeesh Kumar1, Subranshu Sekhar Dash2, M. Arun Bhaskar3, C. Subramani4, and S.Vivek5 1,3,5

Dept. of EEE, Velammal Engineering College, Chennai-66 Dept. of EEE, SRM University, Kattankulathur, Chennai [email protected]

2,4

Abstract. An Unified Power Flow Controller (UPFC) is a typical Flexible AC Transmission System (FACTS) device playing a vital role as a stability aid for large transient disturbances in an interconnected power system. This paper deals with the design of Feedback Linearising Controller (FBLC) tuned by PSO for UPFC.The disturbances are created in the SMIB system and the results are simulated. The results proved the supremacy of the Power System Stabilizer (PSS) equipped with FBLC over the PSS equipped with Proportional Integral (PI) controller. Keywords: UPFC, PSO, FACTS, FBLC, PSS, PI.

1 1.1

Introduction Transient Stability

Transient stability refers to ‘‘the ability of a power system to maintain synchronism when subjected to a severe transient disturbance’’ [1]. Stabilization of a synchronous generator is undoubtedly one of the most important problems in power system control. Power system stabilizers (PSS) and Automatic voltage regulators (AVR) with exciter are normally employed to damp out the electromechanical oscillations as well as for the post fault bus-voltage recovery. However, it is well known that the performances of PSS and AVR are limited since their designs are primarily based on linear control algorithms. In the event of large faults, the nonlinearities of the system become very severe, thereby putting limitations on the performances of classical control designs. Thus, the most appropriate approach for controller design for a power system is the use of nonlinear control theory, i.e., multivariable feedback linearization scheme. The application of feedback linearization approaches for power system control was first investigated by Marino [2] and subsequently by several researchers [3-5]. This control technique has also been successfully applied to control of drives and power electronics based systems [6-8]. Successful applications of FACTS equipment for power flow control, voltage control and transient stability improvement have been reported in the literature [9-13].The rapid development of power electronics technology provides opportunities to develop new power equipment to improve the performance of the actual power systems. During the last decade, a technology called “Flexible AC Transmission Systems” (FACTS) have been proposed and implemented. B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 135–142, 2011. © Springer-Verlag Berlin Heidelberg 2011

136

M. Jagadeesh Kumar et al.

2 Modeling of UPFC 2.1 UPFC Installed in SMIB System The mathematical model of the UPFC is derived here in the d-q (synchronously rotating at the system angular frequency ω) frame of reference. This is followed by a detailed description of the conventional PI control strategy employed for active and reactive power control using UPFC. The equivalent circuit model of a power system equipped with a UPFC is shown in Fig 2. The series and shunt VSIs are represented by controllable voltage sources Vc and Vp, respectively. Rp and Lp represent the resistance and leakage reactance of the shunt transformer respectively. Leakage reactance and resistance of series transformer have been neglected.

Fig. 1. Single line diagram of UPFC installed in power system

2.2 Modeling of Shunt Converter R

L

i R

L

L

V

V

(1)

V

V

(2)

Where: i V

i i i V V V

V

V V V 0 0 0 0

T T T

0 0 and

0 0 0

0 0

0

Under the assumption that the system has no zero sequence components, all currents and voltages can be uniquely represented by equivalent space phasors and then

Comparison of PSO Tuned Feedback Linearisation Controller (FBLC) and PI Controller

137

transformed into the synchronous d-q-o frame by applying the following transformation (q is the angle between the d-axis and reference phase axis): 2 3 2 3

cos 2 3

sin

sin

1

1

√2

√2

2 3 2 3

cos sin 1 √2

Thus, the transformed dynamic equations are given by, R R

L

L

L

L

V

V

ωi

(3)

V

V

ωi

(4)

where ω is the angular frequency of the AC bus voltage.

Fig. 2. PI-Control Structure of Shunt Converter

2.3 Modeling of Series Converter sin

(5)

cos

(6)

For fast voltage control, the net input power should instantaneously meet the charging rate of the capacitor energy. Thus, by power balance,

= Vdcidc

(7)

An appropriate series voltage (both magnitude and phase) should be injected for obtaining the commanded active and reactive power flow in the transmission line, i.e.,

138

M. Jagadeesh Kumar et al.

(Pu,Qu) in this control. The current references are computed from the desired power references and are given by, (8) (9)

The corresponding control system diagram is shown in Fig 6 .

Fig. 3. PI-Control of Series Converter

3 Particle Swarm Optimisation PSO is basically developed through simulation of bird flocking in a two-dimension space. The position of each agent is represented by its XY-axis position and the velocity is expressed by VX (the velocity of x-axis) and VY (the velocity of y-axis). Modification of the agent position is realized by position and velocity information. Bird flocking optimizes a certain object function. Each agent knows its best value (pbest) and its XY position. This information is an analogy of personal experiences of each agent. Moreover, each agent knows the best value in the group (gbest) among bests. This information is an analogy of knowledge of how the other agents around them have performed. Namely, each agent tries to modify its position. • • • •

The current position (X,Y) The current velocities( VX, VY) The distance between the current position and pbest The distance between the current position and gbest

where vi,j velocity of particle i and dimension j, xi,j position of particle i and dimension j c1,c2 Acceleration constants, w inertia weight factor, r1,r2 Random numbers between 0 and 1, pbest Best position of a specific particle, gbest Best particle of the group

Comparison of PSO Tuned Feedback Linearisation Controller (FBLC) and PI Controller

139

4 FBLC for UPFC In this section, the design steps for the feedback linearizing control of UPFC have been presented. In the UPFC control, there are two objectives, i.e. ac bus voltage (Vdc) control. in the following control design, the dynamic equation for Vs is obtained as follows with reference to single line diagram. Now, for the control design, the complete state space model is expressed in the form of equations as follows:

The outputs of the system are Vs and Vdc. thus,

By control law, the outputs of the system is given by, 3 4 6 5

0 0 0 0

0 0

The non-singularity of E(x) can be observed by computing the determinant of E(x). E(x) is non-singular in the operating ranges of Vs and Vdc. For tracking of Vs and Vdc, the new control inputs v1, v2, v3 and v4 are selected as (by both proportional and integral control):

140

M. Jagadeesh Kumar et al.

The gain parameters K11, K12, K21, K22,K23are determined by assigning desired poles on the left half s-plane and, thus, asymptotic tracking control to the reference can be achieved.

5 Simulation Results 5.1 CASE-I The synchronous generator is assumed to operate at P=1.2p.u.and Q = 0.85 p.u. A 3phase fault occurs near the infinite bus for duration of 100 ms The 3-phase fault is created at 500ms and removed at 600ms.

Fig. 4. Comparison of transient responses between FBLC and PI-Controller

Comparison of PSO Tuned Feedback Linearisation Controller (FBLC) and PI Controller

141

5.2 CASE-II The synchronous generator is assumed to operate at P=1.2p.u.and Q = 0.5 p.u. A 3phase fault occurs near the infinite bus for duration of 100 ms The 3-phase fault is created at 500ms and removed at 600ms.

Fig. 5. Comparison of transient responses between FBLC and PI-Controller

6 Conclusion The mathematical model for Unified Power Flow Controller (UPFC) has been developed and the MATLAB experimental results proved the supremacy of the Feed-Back Linearization Controller (FBLC) over the ordinary Proportional Integral (PI) Controller. For all type of disturbances created in the Single Machine Infinite Bus (SMIB) system, the Feed-Back Linearization Controller (FBLC) equipped with Power System Stabilizers (PSS) damped the electromechanical oscillations faster than the Proportional Integral (PI) controller equipped with Power System Stabilizers (PSS).

References [1] Anderson, P.M., Fouad, A.A.: Power System Control and Stability. IEEE Press (1994) [2] Akhrif, O., Okou, F.A., Dessaint, L.A., Champagne, R.: Application of a multivariable feedback linearization scheme for rotor angle stability and voltage regulation of power systems. IEEE Transactions on Power Systems 14(2), 620–628 (1999) [3] Tan, Y.L., Wang, Y.: Design of series and shunt FACTS controller using adaptive nonlinear coordinated design techniques. IEEE Transactions on Power Systems 12(3), 1374–1379 (1997)

142

M. Jagadeesh Kumar et al.

[4] Gyugyi, L.: Dynamic compensation of AC transmission lines by solid-state synchronous voltage sources. IEEE Transactions on Power Delivery 9, 904–911 (1994) [5] Mihaliæ, R., Zunko, P., Papiæ, I., Povh, D.: Improvement of transient stability by insertion of FACTS devices. In: IEEE/NTUA Athens Power Tech. Conference Proc., pp. 521–525 (1993) [6] Saravanailango, G., Nagamani, C.: A non-linear control technique for UPFC based on linearisation. International Journal of Electric Power Components and Systems 36, 432–447 (2008)

A Nelder-Mead PSO Based Approach to Optimal Capacitor Placement in Radial Distribution System Pradeep Kumar and Asheesh K. Singh Electrical Engineering Department, MNNIT Allahabad, India [email protected], [email protected]

Abstract. In distribution system, the size and location of shunt capacitors for peak power loss and energy loss reduction plays a vital role. This paper proposes a new method, Nelder-Mead particle swarm optimization (NM-PSO) for the optimal capacitor placement problem. The NM-PSO method is applied to IEEE 69-bus radial distribution system and the results obtained are compared with that of particle swarm optimization (PSO). Also, the problem has been reformulated to include the maintenance cost and economic factors such as inflation and interest rates. The results obtained clearly indicate the better performance of NM-PSO over PSO, confirming its superiority to PSO in finding the optimal solution and handling more complex, nonlinear objective functions.

1 Introduction The power system aims to feed the customer loads with quality electrical energy as economical as possible. Distribution system, one of the three components of power system is responsible for energy transfer to these electrical loads. Studies indicate that in a power system major portion of the total system losses, nearly 13% occur in distribution system [1]. With development, the increase in quantity of electrical loads will make the system more inefficient by increasing the distribution system losses. Therefore, the utilities need to increase the efficiency of the distribution system. To reduce these power losses the utilities provide the reactive power demands locally by shunt capacitors. Due to their economical cost and high cost benefits shunt capacitors have been the preferred choice of the utilities. However, the amount of the profit that can be achieved using these capacitors largely depends upon their optimal location and size. Investment cost is one of the most important factors for optimal capacitor placement (OCP) amongst other factors such as cost of planning, operation, design process investment, power or energy etc. Thus, economic consideration becomes necessary for obtaining the best solution in capacitor placement design. This problem of OCP has attracted many researchers, taking the problem’s publication count to over 400 [1]. Various heuristic methods [2, 3] and Numerical programming [4] are also used for solving the problem. Recently, artificial intelligence based methods [5-8] have also been used to optimally placing the capacitors. Researchers have modeled the OCP problem as different objective functions. In some papers, maximizing dollar saving function [3], [5-9] remains the objective while in some it remains only reducing the power losses [10]. But these objective functions assume the B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 143–150, 2011. © Springer-Verlag Berlin Heidelberg 2011

144

P. Kumar and A.K. Singh

costs to remain constant over the planning period, without including the maintenance cost, market inflation and creditor’s interest. This paper proposes a new optimization algorithm of evolutionary computation, known as Nelder-Mead particle swarm optimization (NMPSO) [10], for solving OCP problem. NM-PSO combines the advantages of both NelderMead (NM) and particle swarm optimization (PSO) algorithms. A new objective function of the problem with the objectives of cost reduction of peak power loss, energy loss and capacitor’s cost with effects of maintenance cost, inflation and interest rates on cost. The candidate bus selection is done using the bus sensitivity method [12]. The NM-PSO algorithm is applied to the IEEE 69-bus radial distribution system and the results are compared with PSO algorithm. The results obtained show the applicability of NM-PSO over PSO for OCP. The results show that with NM-PSO mores savings can be achieved with less kVArs placed in the system in comparison to PSO.

2 PSO and NM-PSO This section describes the PSO and NM-PSO briefly. The PSO algorithm is an evolutionary computation technique, imitating the social behavior amongst entities. In NM-PSO, the efficient local direct search technique is combined with PSO. This helps PSO is searching the space effectively. 2.1 Method 1: Particle Swarm Optimization (PSO) PSO [13, 14] is an evolutionary computation technique imitating the social behavior of bird flocking and fish schooling. It is a population based stochastic algorithm, where population and individual entities are termed as swarm and particles, respectively. These particles move through the problem hyperspace with given velocity. In PSO, particles in the swarm update their positions stochastically towards the successful regions. The success of particles is influenced by particle’s previous best position and success of their topological neighbors. The detailed analysis of this algorithm is presented in [13, 14]. For an n-dimensional search space the position ( x i (t ) ) and velocity ( v i (t ) ) of particles, at any instant t, can be given as:

xi (t ) = [x1 (t ), x 2 (t ), x 3 (t ),  , x n (t )]T

(1)

v i (t ) = [v1 (t ), v 2 (t ), v 3 (t ),  , v n (t ) ]T

(2)

Keeping the relative importance of particle’s own and neighborhood experiences, the updated velocity of the particle is given as

v i (t + 1) = v i (t ) + ϕ1 .rand 1 .( p best − x i (t ) + ϕ 2 .rand 2 .( g best − x i (t ))

(3)

where, φ1, φ2 are two positive numbers (1.5, 2), rand1, rand2 are two random numbers [0, 1], pbest is local best of particles, gbest is global best of particles. v i (t ) is the velocity of the particles taken as difference between maximum and minimum

A Nelder-Mead PSO Based Approach to Optimal Capacitor Placement

145

capacitor values divided by number of iteration respectively. Accordingly, the updated position of the particle is given as

xi (t + 1) = xi (t ) + vi (t + 1)

(4)

The pseudo code of the above algorithm is shown in Fig.1. 1. Initialize each particle End

2. Do For each particle Calculate fitness value If the fitness value is better than the best fitness value (pl) set current value as the new pl End Choose the particle with the best fitness value of all the particles as the pg For each particle Calculate particle velocity according equation (3) Update particle position according equation (4) End Fig. 1. Pseudo code of particle swarm optimization (PSO)

2.2 Method 2: Nelder-Mead Particle Swarm Optimization (NM-PSO) The hybrid NM-PSO algorithm, as proposed by Fan and Zahara [10], is based on combination of NM method and PSO for constrained optimization. Nelder and Mead proposed a simple local direct search technique, which does not require any derivative for finding solution of any function [15]. The PSO is used as global search technique but it is limited by high computational cost of the slow convergence rate. The slow convergence rate of PSO than NM is due to improper utilization of local information to determine a most promising search direction. To overcome this slow convergence PSO is combined with NM, in a way that both the algorithms enjoy merits of each other. In NM-PSO, PSO prevents the hybrid approach from getting trapped in local optima, whereas NM increases the convergence rate. Summarily, PSO focuses on “exploration” and NM method focuses on “exploitation” [11]. For an N-dimensional problem, the pseudo code of the algorithm is given in Fig. 2. 1. 2. 3. 4.

Initialize. Generate a population of size 3N+1 Repeat Evaluate the fitness of each particle Rank them based on the fitness results. Simplex Method. Apply NM operator to top N+1 particle. PSO method. Apply PSO operator to remaining 2N particles, and update the particles with worst fitness until stopping criterion is reached

Fig. 2. Pseudo code for Nelder-Mead particle swarm optimization (NM-PSO)

146

P. Kumar and A.K. Singh

There are various parameters to be specified for the experiments. The parameters for PSO have already been discussed in previous sub-section. The various parameters of the NM algorithm in NM-PSO viz. reflection coefficient (α), expansion coefficient (β), Contraction coefficient (γ), and Shrink Coefficient (δ) are given as 1, 2, 0.5, and 0.5, respectively. The experiments are performed for 100 iterations.

3 Bus Sensitivity Sensitivity of any particular bus can be calculated using bus sensitivity method [12], which reveals that reactive power compensation is mainly required on buses having reactive power demand and below nominal bus voltage. The index bus sensitivity for each bus i bus sensitivity (BSi) index is defined as

BS i = λQi

Ql i Vi

(5)

where, λQi, Qli, and Vi are the Lagrange multiplier, load reactive power and voltage magnitude at bus i, in that order. For the capacitor placement problem, λQi imitates the sensitivity of objective function towards the changes in the reactive power injection to bus bar i. Bus sensitivity index BSi gives cost required to increase the voltage at bus i. The unit of BSi is $/V.

4 Problem Formulation The objectives of OCP are to minimize power loss, energy loss, and cost of capacitors, along with maintaining the bus voltages in a limit. The costs included in the objective function are affected by the maintenance cost and the economic factors. The reformulated objective function incorporating these costs and factors, can be expressed as nl

nc

i =1

j =1

min f (u ) = k e  Ti Pi + k p Pl +  0

( )

C j u 0j

k efY

Lc

(

+  M tc (1 + I )Y −1 Y =1

)

(6)

where, Ti is time duration, nl is load level, Cj(u0j) is cost function of capacitor

C j (u 0j ) = kinst + kcj u 0j , nc is number of capacitors, Pl is power loss at peak load, kef is economic factor given as k ef =

1+ m , and Pi power loss at load level i. The 1+ f

values for capacitor cost (kcj), capacitor installation cost (kinst), cost of energy (ke), cost of peak power loss (kp), Planning period (Y), maintenance cost (Mtc), increment (I) in Mtc, inflation rate (m), and interest rate (f ) are taken as US$ 3/kVAr, US$ 1000, US$ 0.06/kWh, US$ 168/kW, 10 years, US$ 322.42/year, 10% per year, 4% per year, and 5% per year, respectively. Initially, based on distribution load flow [16], the power losses for the system are calculated without any capacitor placement. BSi calculation forms the basis of bus

A Nelder-Mead PSO Based Approach to Optimal Capacitor Placement

147

selection for capacitor placement. Then at the selected buses the capacitors are placed utilizing the two methods (i.e. PSO and NM-PSO) and the objective function. For the given system the process of determination of the OCP is done till the minimum value of the objective function is obtained following the constraints. Performance of the system is then evaluated based on the solutions obtained. 4.1 Operating Constraints The problem of capacitor placement is subjected to the following constraints: (1). AC power flow constraints. The power flow constraint as be expressed as:

G i ( x i , u i ) = 0;

i = 1,2,..., nl

(7)

(2). Bus voltage constraint. The bus voltages should lie in between the minimum, Vmin (0.9 pu) and Vmax (1.1 pu).

Vmin < V < Vmax

(8)

(3). Capacitor capacity constraint. The maximum number of capacitors to be installed in a system is limited, i.e.

u 0j us

∈ {1,2,  , nc}

(9)

5 Results and Discussions The proposed objective function and algorithm, has been implemented on the MATLAB platform. It has been implemented on 12.6 kV IEEE 69-bus system [17] radial distribution system. The diagram of the test feeder system is shown in Fig. 3. Load duration data for the IEEE 69-bus system is given in Table 1. 5.1 Candidate Bus Selection Bus sensitivity method is adopted for candidate bus selection. Buses having highest numeric value of BSi are selected for capacitor placement. The bus sensitivity calculation for IEEE 69-bus system, on this basis 9 buses are selected for capacitor placement, i.e. 11, 12, 21, 48, 49, 50, 59, 61, and 64. Based on the capacitors selected the problem of OCP becomes a 9 (N=9) dimensional problem. 5.2 Capacitor Placement at Buses The capacity of capacitors placed on selected buses of test system is calculated using method-1 and method-2. Table 2 shows the capacitors placed at the candidate buses using the two methods. After placement of capacitors the performance of the bus

148

P. Kumar and A.K. Singh

Fig. 3. IEEE 69-bus radial distribution system

Fig. 4. Bus sensitivity of IEEE 69-bus radial distribution system

Table 1. Load Duration Data for IEEE 69-bus System

Load Level Time Interval (Hrs)

Low 0.5 1000

Medium 1 6760

Peak 1.4 1000

systems is evaluated by considering peak power loss, energy loss, total cost and the benefits. The results obtained are tabulated in Table 3. From Table 2 and 3 it is clearly visible that with method-1 the OCP results in considerable reduction in peak power loss of 29.58%. Also, the cost energy losses are reduces to US$ 220,445.3 from the original US$ 305513.9. This results in yearly benefits of US$ 95, 820.17, which is nearly 19.22%. Method-2 applied for capacitor placement also provides 29.58% peak power loss reduction, but the cost of energy losses reduces from US$ 220445.30 to US$ 220408 in comparison to method-1. The yearly benefit of US$ 99914.11(20.04%) is achieved with method-2 which is higher than that obtained with method-1. Although, the results obtained for both the methods are comparable in terms of losses, but the kVArs placed by the two methods makes the difference. Table 2 shows that for the given system with method-1 15,450 kVArs are placed, whereas with method-2 only 14,100 kVArs are placed. This indicates that for OCP problem, NMPSO provides more efficient solution than PSO. Table 2. Capacitors Placed at the Candidate Buses

Bus Method 1 Method 2

11 2400 2100

12 2700 2400

21 900 900

48 3450 2400

49 600 750

50 1200 1200

59 450 750

61 3000 2700

64 750 900

Total 15,450 14,100

A Nelder-Mead PSO Based Approach to Optimal Capacitor Placement

149

Table 3. Capacitors Placed at the Candidate Buses

Peak Power Loss (kW) Medium Power Loss (kW) Low Power Loss (kW) Total Power Loss (kW) % Peak Power Loss Reduction % Loss Reduction Cost of Peak Power Loss (US$) Cost of Energy Loss (US$) Capacitor Cost (US$) Total Cost (US$) Benefits (US$) % Benefits

Original Configuration 1149 563.34 134.72 1847.06 193032 305513.9 498545.9 -

Method 1 809.11 397.51 177.81 1384.43 29.58 25.05 135930.50 220445.30 46350 402725.70 95820.17 19.22

Method 2 809.07 397.47 177.5 1384.04 29.58 25.07 135923.80 220408 42300 398631.80 99914.11 20.04

6 Conclusion In this paper, the proposed NM-PSO based algorithm is tested on IEEE 69-bus radial distribution system for OCP. To make the problem more realistic, objectives of the problem were modified to include the reduction in total cost and real power losses, with inclusion of economic factor and maintenance cost. The work demonstrates that the inclusion of variation in the costs calculated in the objective function gives the problem a more realistic approach. The computational effort for NM-PSO is balanced by dividing the total number population size (3N+1) into NM (N+ 1 particle) and PSO (2N particles). NM being inherently fast in computes the local solutions while the global solutions are searched by the PSO. The results presented help in establishing the NM-PSO as a promising and viable tool for finding the optimal solution and handling more complex, nonlinear objective functions.

References 1. Ng, H.N., Salama, M.M.A., Chikani, A.Y.: Classification of Capacitor Allocation Techniques. IEEE Trans. Power Del. 15(1), 387–392 (2000) 2. Baran, M.E., Wu, F.F.: Optimal Capacitor Placement on Radial Distribution Systems. IEEE Trans. Power Del. 4(1), 725–734 (1989) 3. Baldick, R., Wu, F.F.: Efficient Integer Optimization Algorithms for Optimal Coordination of Capacitors and Regulators. IEEE Trans. Power Syst. 5(3), 805–812 (1990) 4. Abdel-Salam, T.S., Chikhani, A.Y., Hackam, R.: A New Technique for Loss Reduction Using Compensating Capacitors Applied to Distribution Systems with Varying Load Condition. IEEE Trans. Power Del. 9(2), 819–827 (1994) 5. Sundhararajan, S., Pahwa, A.: Optimal Selection of Capacitors for Radial Distribution Systems using a Genetic Algorithm. IEEE Trans. Power Syst. 9(3), 1499–1507 (1994) 6. Salama, M.M.A., Chikhani, A.Y.: An Expert System for Reactive Power Control of a Distribution System Part 1: System Configuration. IEEE Trans. Power Del. 7(2), 940–945 (1992)

150

P. Kumar and A.K. Singh

7. Ng, H.N., Salama, M.M.A., Chikhani, A.Y.: Capacitor Allocation by Approximate Reasoning: Fuzzy Capacitor Placement. IEEE Trans. Power Del. 15(1), 393–398 (2000) 8. Chang, C.F.: Reconfiguration and Capacitor Placement for Loss Reduction of Distribution Systems by Ant Colony Search Algorithm. IEEE Trans. Power Syst. 23(4), 1747–1755 (2008) 9. Mekhamer, S.F., Soliman, S.A., Moustafa, M.A., El-Hawary, M.E.: Application of Fuzzy Logic for Reactive Power Compensation of Radial Distribution Feeders. IEEE Trans. Power Syst. 18(1), 206–213 (2003) 10. Fan, S.S., Zahara, E.: A Hybrid Simplex Search and Particle Swarm Optimization for Unconstrained Optimization. Eur. Jr. Operational Research 181, 527–548 (2007) 11. Zahara, E., Kao, Y.: Hybrid Nelder-Mead Simplex Search and Particle Swarm Optimization for Constrained Engineering Design Problems. Expert Syst. with App. 36, 3880–3886 (2009) 12. da Silva, I.C., Carneiro, S., de Oliveira, E.J., de Souza Costa, J., Pereira, J.L.R., Garcia, P.A.N.: A Heuristic Constructive Algorithm for Capacitor Placement on Distribution Systems. IEEE Trans. Power Syst. 23(4), 1619–1626 (2008) 13. Abido, A.: Optimal Power Flow Using Particle Swarm optimization. Int. Jr. Elect. Power Energy Syst. 24(7), 563–571 (2002) 14. Abido, A.: Particle Swarm Optimization for Multimachine power System Stabilizer Design. Power Engg. Scoiety Summer Meeting (PES) 3, 1346–1351 (2001) 15. Nelder, J.A., Mead, R.: A Simplex Method for Function Minimization. Comp. Jr. 7, 308– 313 (1965) 16. Cheng, C.S., Shiromohammadi, D.: A Three-phase power Flow Method for Real-Time Distribution System Analysis. IEEE Trans. Power Syst. 10(2), 671–679 (1995) 17. Huang, S.-J.: An Immune Based Optimization Method to Capacitor Placement in a Radial Distribution System. IEEE Trans. Power Del. 15(2), 744–749 (2000)

Comparative Performance Study of Genetic Algorithm and Particle Swarm Optimization Applied on Off-grid Renewable Hybrid Energy System Bhimsen Tudu, Sibsankar Majumder, Kamal K. Mandal, and Niladri Chakraborty Power Engineering Department, Jadavpur University, Kolkata: 700098, India [email protected], [email protected], [email protected], [email protected]

Abstract. This paper focuses on unit sizing of stand-alone hybrid energy system using Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) and comparative performance study of these two meta-heuristic techniques on hybrid energy system. The hybrid system is designed focusing on the viability and combining different renewable energy sources like wind turbines, solar panels along with micro-hydro plant as well as fuel cells to compensate the deficit generation in different hours. Apart from the non-conventional sources, the system has been optimized with converters, electrolyzers and hydrogen tanks. Net present cost (NPC), cost of energy (COE) and generation cost (GC) for power generation have been considered while optimal unit sizing of the system are performed. Feasibility of the system is made based on net present cost (NPC). The performances of two algorithms have been checked for different values of variants of the respective algorithms and a comparative study has been carried out based on number of iterations taken to find optimal solution, CPU utilization time and also quality of solutions. The comparative analysis shows that the Particle Swarm Optimization technique performs better than Genetic Algorithm when applied for the sizing problem. Keywords: Genetic Algorithm, Particle Swarm Optimization, Off-grid renewable energy system, Solar panels, Wind turbine, Micro-hydro plant.

1 Introduction Fast depletion of fossils fuels, direct and indirect green house gases emission from thermal power plants and environmental issues have shifted world’s focus towards the new and renewable sources of energy. Huge amount of electricity is being produced to meet the high energy demand in India and is in fifth position in electricity generation (922.2TWh) in the world at the end of year 2010 [1]. Still many remote B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 151–158, 2011. © Springer-Verlag Berlin Heidelberg 2011

152

B. Tudu et al.

villages of India are waiting for the electricity to taste. It is now of great concern for government to provide power to a remote location where power can’t be supplied from the main grid line due to the high associated cost. This has explored a new horizon and opened up the door of off-grid renewable energy system. For better understanding of performance and proper optimization of this type of hybrid system, different meta-heuristic techniques have been used extensively such as Genetic Algorithm [2-4], Particle Swarm Optimization [5-7], Fuzzy Control, Differential Evolutionary Algorithm, Ant Colony, Tabu Search etc. [8-11].

2 Hybrid Test System Model The hybrid test system model has been framed keeping in mind a typical village of India. It is assumed that the village has high potential of solar energy, wind energy along with the micro hydro plant. The sizing of the proposed hybrid energy system has been carried out considering an increase of 10% in average load profile over the lifetime of the project of 30 years. The average hourly wind velocity and solar insolation data and load profile for 24 hours of that location are shown in Figure 1. The different components used in this system are standard components of different manufacturer make and data used for different components are listed in Table 1.

Fig. 1. Hourly load, solar insolation and wind velocity data for the site

Comparative Performance Study of Genetic Algorithm and Particle Swarm Optimization

153

Table 1. Different parameter values & costs (Rs./Yr) considered for the hybrid energy system Component Parameter

Value

Capital Cost

Micro Installed capacity 20kW Hydro-plant Operating efficiency 50% lifetime 20Years Wind Capacity 10kW Turbine Rated speed 12m/s Cut–in speed 2.5m/s Furling 15.6m/s speed Lifetime 15years Solar PV Rated power 0.15kW Module Slope 25 degrees Derating Factor 0.9 Lifetime 20 years Fuel Cell Capacity 5kW Efficiency 50% Lifetime 15Years Electrolyzer Capacity 1kW Efficiency 90% Lifetime 20 Years Converter Capacity 5kW Efficiency 90% Lifetime 20 Years Hydrogen Capacity 1 Kg Tank Lifetime 15 Years

Replacement Operation Cost Cost

1610K

1288K

32K

1334K

1104K

46K

23K

18.4K

NIL

552K

460K

23K

92K

82K

9.2K

115K

92K

0.92K

55.2K

46K

0.46K

3 System Costs and Objective Function The total annualized cost of a component consists of the annualized capital cost, annualized replacement cost, annualized operation and maintenance cost as well as annualized fuel cost. The total annualized cost of different components of the hybrid systems can be represented by the equations shown below [12]. Total annualized cost of each component: C ann, component

= C acap, component

+

C arep , component

+

C aom, component

C afc , component

+

(1)

Where, C acap = annualized capital cost, C arep = annualized replacement cost, C aom = annualized operation & maintenance cost, C afc = annualized fuel cost.

154

B. Tudu et al.

The total annualized cost of the hybrid energy system can be expressed as [12]:

=

C ann, tot +

Ns



C ann,

s =1

solar , s

+

Nw



C ann,

w =1

wind , w

Nt

Ne

Nf

t =1

e =1

f =1

+

Nc

 C ann, converter, c

c =1

 C ann, tan k , t +  C ann, electo, e +  C ann, fuelcell, f

(2)

+ C ann,

hydro

Where, N s = number of installed solar modules, N w = number of installed wind turbines, N c = number of installed converters, N t = number of installed hydrogen tanks, N e = number of installed electrolyzers and N f = number of fuel cells installed in the hybrid system. The net present cost of the system is expressed as: C NPC

=

C ann , tot CRF proj

(3)

The cost of energy of the system is given by [13] as: COE

=

C ann ,

tot

Total load catered by the system over the year in kWh

(4)

Similarly the generation cost of the system can be expressed as: GC =

C ann , tot Total power generated over the year in kWh

(5)

As in some hours of operation, the total power generated by the system is not completely utilized by the demand side and there is some input to the electrolysers, the value of COE and GC differs by some extent. In this study, the system is optimized based on the net present cost.

4 Optimization Methods The objective function derived above for obtaining the optimal sizing of the different components of the hybrid energy system under study has been optimized employing Particle Swarm Optimization technique and Genetic Algorithm. 4.1 Particle Swarm Optimization Particle swarm optimization was originally developed by Kenedy and Eberhart [14] based on natural behaviour of the particles or agents and it is quite efficient population based optimization procedure for multidimensional non-linear complex problem. Each position of the particle in d-dimensional search space represents the possible solution of the problem. After each iteration, particles change their position

Comparative Performance Study of Genetic Algorithm and Particle Swarm Optimization

155

to give the better solution. Each time particles update their position in the search space depending upon their own experience and their neighbour particles’ experience. A particle’s position is updated depending upon its current velocity, its own distance from its previous best position and distance from the best position occupied by the particle in the group. According to Kennedy et al., for each particle i, its position is updated in the following manner:

X ki + 1 = X ki + V ki + 1 Where

(6)

Vki+1 is pseudo velocity and can be calculated as follows: Vki +1 = wk Vki + c1r1 ( Pki − X ki ) + c2 r2 ( Pkg − X ki )

Here k represents a pseudo time increment, particle i at time k and

(7)

i k

P represents the best ever position of

g k

P represents the global best position in the group of swarm.

r1 & r2 represent the uniform random values in the range [0 1], w represents the weight parameters and c1 & c2 are the acceleration constant which depend on the cognitive and social nature of the swarms. 4.2 Genetic Algorithm Genetic algorithm is started with a random initial population of chromosomes. The fitness function is evaluated for each chromosome of the corresponding population. A selection operation is performed on the population of chromosomes which will be subjected to crossover and mutation operations based on Roulette Wheel method [2] to produce the new generation. The minimum value of objective function found for a generation of chromosomes is compared with the minimal value found in the previous generation. If the current minimal value is lower than that of the previous generation, then this value is considered to be the optimal solution of the minimization problem. The optimal chromosome found in each generation represents the sizing of different components for the hybrid energy system. From one generation to next generation four steps are involved: a)

Selection: selection of individuals (chromosomes) for reproduction according to their fitness (objective function value) b) Crossover: merging the genetic information of two individuals based on crossover probability factor. c) Mutation: a random alternation of the chromosomes based on mutation probability factor. The positive effect is preservation of genetic diversity and as an effect that local maxima can be avoided. d) Sampling: procedures which computes a new generation from the previous one and its off-spring.

156

B. Tudu et al.

5 Simulated Results and Comparative Study The objective function formulated for obtaining the sizing of the hybrid energy system is optimized using both Particle Swarm Optimization and Genetic Algorithm. A comparative analysis is performed varying the values of different optimization variants to find their best fitted values for the objective function under study. The optimal sizing as found employing Genetic Algorithm and Particle Swarm Optimization is given in Table 2. We performed the analysis taking the different values of acceleration constant and swarm population for Particle Swarm Optimization technique keeping weight parameter constant. We considered different values for acceleration constant in the range [0.1 2]. Similarly for swarm population, different values in the range [10 200] were considered. Based on our analysis, we found that the technique performs best when acceleration constant is 2 and the swarm population is 150. Table 2. Optimal sizing obtained for the integrated hydro-wind-solar-fuel cell system Component

Optimal Sizing (No. of Units) Wind Turbine 8 Solar Panel 0 Converter 3 Fuel Cell 3 Electrolyser 5 Hydrogen Tank 3 Net Present Cost (NPC): Rs. 24327.45k Generation Cost (GC): Rs. 4.98/kWh Cost of Energy (COE): Rs. 7.53/kWh

Capacity 80 kW 0 kW 15 kW 15 kW 5 kW 3 kg.

Again, we performed similar analysis for Genetic Algorithm varying the values of crossover probability factor, mutation probability factor and chromosome population. We considered different values for crossover and mutation probability factors in the range [0.1 0.9]. In the same way, for chromosome population, different values in the range [10 200] were considered. Analyzing the performance of Genetic Algorithm in different runs, it was found that it performs best for the objective function when crossover and mutation probabilities are 0.5 and chromosome population is 200. Using the fine tuned values obtained by varying the parameters of PSO and GA, the comparative analysis between the two optimization techniques was performed. The comparison of performance was mainly based on the number of iterations needed to reach the optimal value, CPU utilization time as well as quality of solutions. A sampling of 100 separate runs was taken for each of the techniques. The performances of these two optimization techniques in terms of average number of iteration needed to find the optimal value and the average CPU Utilization time are given in Table 3.

Comparative Performance Study of Genetic Algorithm and Particle Swarm Optimization

157

Table 3. Performance comparison of PSO and GA Method PSO GA

Avg. No. Of Iteration to Find the Optimal Value 2 13

CPU Utilization Time (Sec.) 12.7 108.5

Figure 2 shows the number of iteration to reach the optimal value with the fine tuned values of the variants for Genetic Algorithm and Particle Swarm Optimization.

Fig. 2. The optimal Net Present Cost (NPC) in terms of the iterations for GA & PSO

5 Conclusion From the analysis, it can be derived that PSO performs a way ahead of Genetic Algorithm both in terms of minimum number of iterations needed to reach the optimum as well as CPU utilization time period. When analysis was performed for samples of 100 different runs, we found that PSO reaches the optimal value using 85% less iterations and consuming 88% less CPU time. So PSO is a more preferred technique of optimization rather than Genetic Algorithm for our test system.

References 1. BP Statistical Review of World Energy (June 2011), http://www.bp.com 2. Koutroulis, E., Kolokotsa, D., Potirakis, A., Kostas, K.: Methodology for optimal sizing of stand-alone photovoltaic/wind-generator systems using genetic algorithms. Solar Energy 80, 1072–1088 (2006) 3. Dufo-Lopez, R., Bernal-Agustin, J.L.: Design and control strategies of PV-Diesel systems using genetic algorithms. Solar Energy 79, 33–46 (2005)

158

B. Tudu et al.

4. Ould Bilal, B., Sambou, V., Ndiaye, P.A., Kebe, C.M.F., Ndongo, M.: Optimal design of a hybrid solar-wind-battery system using the minimization of the annualized cost system and minimization of the loss of power supply probability (LPSP). Technical note, Renewable Energy 35, 2388–2390 (2010) 5. Hakimi, S.M., Moghaddas-Tafreshi, S.M.: Optimal sizing of a stand-alone hybrid power system via particle swarm optimization for Kahnouj area in south-east of Iran. Renewable Energy 34, 1855–1862 (2009) 6. Hakimi, S.M., Tafreshi, S.M., Kashefi, A.: Unit sizing of a stand-alone hybrid power system using particle swarm optimization (PSO). In: Proceedings of the IEEE International Conference on Automation and Logistics, Jinan, China, August 18-21 (2007) 7. Hakimi, S.M., Tafreshi, S.M., Rajati, M.R.: Unit sizing of a stand-alone hybrid power system using model free optimization. In: 2007 IEEE International Conference on Granular Computing (2007) 8. Li, M., Wang, C.: Research on optimization of wind and PV hybrid power systems. In: Proceedings of the World Congress on Intelligent Control and Automation, Chongqing, China, June 25-27 (2008) 9. Qi, Y., Jianhua, Z., Zifa, L., Shu, X., Weiguo, L.: A new methodology for optimizing the size of hybrid PV/wind system. In: ICSET (2008) 10. Xu, D., Kang, L., Cao, B.: Graph-Based Ant System for Optimal Sizing of Standalone Hybrid Wind/PV Power Systems. In: Huang, D.-S., Li, K., Irwin, G.W. (eds.) ICIC 2006. LNCS (LNAI), vol. 4114, pp. 1136–1146. Springer, Heidelberg (2006) 11. Katsigiannis, Y.A., Georgilakis, P.S.: Optimal sizing of small isolated hybrid power systems using tabu search. Journal of Optoelectronics and Advanced Materials 10(5), 1241–1245 (2008) 12. Dufo-Lopez, R., Bernal-Agustin, J.L.: Design and control strategies of PV-Diesel systems using genetic algorithms. Solar Energy 79, 33–46 (2005) 13. Katsigiannis, Y.A., Georgilakis, P.S., Karapidakis, E.S.: Genetic Algorithm Solution to Optimal Sizing Problem of Small Autonomous Hybrid Power Systems. In: Konstantopoulos, S., Perantonis, S., Karkaletsis, V., Spyropoulos, C.D., Vouros, G. (eds.) SETN 2010. LNCS, vol. 6040, pp. 327–332. Springer, Heidelberg (2010) 14. Kennedy, J., Eberhart, R.: Particle swarm optimization. In: Proceedings of the IEEE International Conference on Neural Networks IV, pp. 1942–1948 (1995)

An Efficient Algorithm for Multi-focus Image Fusion Using PSO-ICA Sanjay Agrawal, Rutuparna Panda, and Lingaraj Dora Department of Electronics & Telecommunication Engineering, VSS University of Technology, Burla, Odisha, India [email protected]

Abstract. A pixel-level multi-focus image fusion scheme using Independent Component Analysis (ICA) and Particle Swarm Optimization (PSO) is proposed in this paper. The novelty in this work is optimization of ICA bases using PSO and its application to multi-focus image fusion, which is not found in the literature. The idea is to divide the input registered images into patches and get the independent components using ICA transform. The images are then fused in the transform domain using pixel-based fusion rules. PSO is used to optimize the independent components in ICA. We observe that the proposed method outperforms the existing fusion techniques using ICA.

1 Introduction The process of combining multiple images of the same object into a single image is known as Image Fusion. A wide variety of data acquisition devices are available at present. Hence, the scope of image fusion has increased manifold and it has become an important area of research. There are now sensors available which cannot generate images of all objects at various distances (from the sensor) with equal clarity (e.g. camera with finite depth of field, light optical microscope and so on). Thus several images of an object are captured, with focus on different parts of it. The captured images are complementary in many ways and a single one of them is not sufficient in terms of their respective information content. The advantage of multi-focus data can be fully exploited by integrating the sharply focused regions seen in the different images using suitable image fusion techniques [1-3]. Multi-focus image fusion techniques have been widely used in the field of image analysis tasks such as target recognition, remote sensing and medical imaging. A number of multi-focus image fusion techniques are seen in the literature. The techniques in which the fusion operation is performed directly on the source images (e.g. weighted- average method) often have serious side effects like reduction of visual perception of the fused image. Other approaches include, image fusion using controllable camera [4], probabilistic methods [5], image gradient method with majority filtering [6], multi-scale methods [7] and multi-resolution methods [8, 9]. Maurer et al. [10] described methods depending on controlled camera motion but that do not work for arbitrary set of images. Probabilistic techniques involve huge computation using floating point arithmetic and thus require a lot of time and memory-space. Image gradient method with majority filtering has the drawback that the defocused B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 159–166, 2011. © Springer-Verlag Berlin Heidelberg 2011

160

S. Agrawal, R. Panda, and L. Dora

zone of one image is enhanced at the expense of focused zone of others. Multiscale methods involve processing and storing of scaled data at various levels which are of same size as that of the original images resulting in a huge amount of memory and time requirement. In this paper, the use of PSO in optimizing the independent components in ICA for a pixel-based multi-focus image fusion algorithm is introduced. The idea is to divide the source images into patches and then express the patches as a linear combination of a set of basis images. It is assumed that the input images have negligible registration problem. Then an analysis kernel and a synthesis kernel is estimated. The transform projects the observed signal on a set of basis vectors. The estimation of these vectors is performed using a population of training image patches and a criterion (cost function), which is going to be optimized using PSO in order to select the basis vectors. First PCA is used to identify the uncorrelated components and then statistically independent vectors are identified using ICA. It is here only PSO is used to obtain and optimize the independent bases which have been identified by using FastICA method [12]. The advantage of our method is in using PSO to optimize the independent components than using a random approach. And the resultant fused images are both qualitatively and visually superior to that of FastICA. The rest of this paper is organized as follows. Section 2 gives the introduction of Independent Component Analysis (ICA). Section 3 introduces the framework of PSO. The proposed fusion scheme using PSO-ICA is described in Section 4. Section 5 gives simulation results and Section 6 gives the conclusion.

2 Independent Component Analysis The image fusion process can be performed at different levels i.e. signal, pixel, feature, and symbolic level of information representation. Nikolov et al. [15] proposed a classification of image fusion algorithms into spatial domain and transform domain techniques. The transform domain image fusion scheme consists of obtaining a transform on each input image and, following specific fusion rules, combining them into a composite transform domain representation. The composite output image is obtained by applying the inverse transform on this composite transform domain representation. Instead of using a standard bases system, such as the DFT, the mother wavelet or cosine bases of the DCT, a set of bases that are suitable for a specific type of image can be trained. A training set of image patches, which are acquired randomly from images of similar content, can be used to train a set of statistically independent bases. This is known as independent component analysis (ICA) [12]. In order to obtain a set of statistically independent bases for image fusion in the ICA domain, training is performed with predefined set of images. Training images are selected in such a way that the content and statistical properties are similar for the training images and the images to be fused. An input image I(x,y) is randomly windowed using a rectangular window W of size NxN, centered around the pixel (mo,no). The result of windowing is an image patch which is defined as [14] Ip(m,n) = W(m,n) * I (m0 - N/2 + m, n0 – N/2 + n) . where m and n take integer values from the interval [0,N-1].

(1)

An Efficient Algorithm for Multi-focus Image Fusion Using PSO-ICA

161

Each image patch Ip(m,n) can be represented by a linear combination of a set of M basis patches bi(m,n), M

I p (m, n) =  vi bi (m, n) i =1

(2) .

where v1,v2,… vM stands for the projections of the original image patch on the basis patch i.e. v i = I p (m, n), b i (m, n) . 2-D representation of the image patches can be simplified to a 1-D representation, using lexicographic ordering. This implies that an image patch Ip (m, n) is reshaped into a vector Ip, mapping all the elements from the image patch matrix to the vector in a row-wise or column-wise fashion. Decomposition of image patches into a linear combination of basis patches can then be expressed as follows:

v1 (t)  v (t)  M I p (t) =  v i (t)b i = [b1 , b 2 ............b M ] *  2  . ........  i =1   v M (t) 

(3)

where t represents the image patch index. Let B = [ b1 , b 2 ............b M ] and V(t) = [v1,v2………..vM]T then equation (3) can be represented as Ip(t) = B*V(t).

(4)

V(t) = B-1*Ip(t) = A*Ip(t) .

(5)

T

Thus, here A = [ a1, a2…..aM ] represents an unknown mixing matrix called analysis kernel and B represents the un-mixing matrix called synthesis kernel. This “transform” projects the observed signal Ip (t) on a set of basis vectors bi. So in the first stage, PCA is used for dimension reduction. This is obtained by eigen value decomposition of the data correlation matrix C = E {Ip(t)Ip(t)T}. The eigen values of the correlation matrix illustrate the significance of their corresponding basis image patches [13]. If V is the obtained K x N2 PCA matrix, the input image patches are transformed by Z (t) =V*Ip (t).

(6)

After the PCA preprocessing step, the statistically independent vectors are selected by using approximations of negentropy by minimizing the objective function JG (w) = [E {G (wTz)}-E {G (v)}] 2

.

(7) T

2

where w is a m-dimensional (weight) vector constrained so that E{(w x) }=1, E{.} represents the expectation operator, ‘v’ is a Gaussian variable with zero mean and unit variance and G(.) is a non-quadratic function. The above function computes one independent component. But for several independent components the following optimization problem is solved using PSO.

162

S. Agrawal, R. Panda, and L. Dora

minimize

wrt. wi, i=1,…,n .

(8)

under constraint E{( x) ( x}= δjk. After the input image patches Ip(t) are transformed to their ICA domain representations V(t), image fusion can be performed in the ICA domain. The equivalent vectors Vk(t) from each image are combined in the ICA domain to obtain a new image Vf(t) as Vf(t) = g(V1(t),…,Vk(t)).

(9)

where g(*) represents the fusion rule. Many fusion rules have been proposed earlier and for multi-focus image fusion, “max-abs” rule i.e. fusion by absolute maximum by selecting the greatest in absolute value of the corresponding coefficients in each image is chosen as Vf(t) = sgn(Vk(t)) max |Vk(t)|. k

(20)

After the composite image Vf(t) is constructed in the ICA domain, the spatial domain representation can be obtained by using the synthesis kernel B, and synthesize the image If (x,y) = B*Vf(t).

(11)

3 Particle Swarm Optimization The particle swarm optimization is an evolutionary computation technique which is a population-based optimization tool that was proposed by Eberhart and Kennedy [16] in 1995. Let N denote the number of particles in a swarm. In general, there are three attributes of a particle, the particle’s current position xi, current velocity vi and local best position pi, in the search space to present their features. Each particle in the swarm is iteratively updated according to the aforementioned attributes. During the search process, the particle successively adjusts its position towards the global optimum according to the two factors: the best position encountered by itself (pbest) which is denoted as pi, j = ( pi1, pi2 ,..., piD ) and the best position encountered by the whole swarm (gbest) which is denoted as pg = ( pg1, pg2,…..,pgD ). Assuming that the objective function f is to be minimized (or maximized), the new velocity of every particle is updated as given below: vi,j(g+1) = w*vi,j(g) + c1*rand1(g)*[pbesti,j(g)-xi,j(g)] + c2*rand2(g)*[gbesti,j(g)-xi,j(g)] for all jЄ1….N.

(12)

vi,j is the velocity of the jth dimension of the ith particle, w denotes the inertia weight of velocity, c1 and c2 denotes acceleration coefficients namely cognitive and social parameter respectively, rand1 and rand2 represent two uniform random functions in the range (0,1) to effect the stochastic nature of the algorithm and g is the number of generations. The new position of a particle is calculated as xi,j(g+1) = xi,j(g) + vi,j(g+1).

(13)

An Efficient Algorithm for Multi-focus Image Fusion Using PSO-ICA

163

4 Proposed PSO-ICA Algorithm for Multi-focus Image Fusion The algorithm of the proposed image fusion method using PSO-ICA is as follows. A set of images with similar content is chosen as the input images. We assume the input images have negligible registration problem. The steps followed in the proposed method are: Step 1 decomposes the input registered images into non overlapping square patches using equation (1). The acquired patches are then ordered lexicographically using equation (3). Step 2 applies the PSO-ICA transform on individual input image patches and converts the images from spatial domain to transform domain using equation (6). We have used the optimization of negentropy as a non- Gaussianity measurement to identify the independent components as explained in [13]. Step 3 uses the max-abs fusion rule by combining the corresponding coefficients from each transformed image to get the fused image using equation (10). Step 4 uses the inverse transformation to convert the fused image into the spatial domain using equation (5). PSO has been used in this paper to minimize the objective function as given in equation (8). The steps followed for PSO-ICA are as follows: • The parameters for PSO are initialized including number of particles in the swarm N, acceleration coefficients c1, c2, inertia weight w and the maximum iterative count. • X, v, pbest are initialized for each particle and gbest for the swarm. • The linear matrix w is calculated for each particle using equation (9). • The fitness value is then calculated for each particle. • Pbest and Gbest is calculated for each particle and the swarm respectively. • The velocity and the position for each particle is updated using equation (12) and (13) The termination condition in the proposed method is the number of iterations or no improvement in ‘gbest’ in a number of iterations.

5 Simulation Results In order to demonstrate the effectiveness of the proposed method, three artificially distorted aircraft images are taken as presented in fig.1. Because a reference image is available, numerical evaluation of the fusion schemes is presented in Table 1. The Signal to Noise Ratio (SNR) expression to compare the reference image with the fused image is:

SNR (dB)

 m  n I ref (m, n) 2 . = 10log10  m  n (I ref (m, n) − I f (m, n)) 2

(14)

The expression for Image Quality Index (Q0) as a performance measure is :

Q0 =

4σ I ref If m Iref m If (m

2 I ref

+ m 2If )(σ 2Iref + σ 2If )

.

(15)

164

S. Agrawal, R. Panda, and L. Dora

where

σ I2 = σ IJ =

M1 M 2 1 ( I (m, n) − m I ) 2  M 1 M 2 − 1 m =1 n =1

M1 M 2 1  ( I (m, n) − mI )( J (m, n) − mJ ) M 1 M 2 − 1 m =1 n =1

mI represents mean of image I(m,n), M1, M2 represent size of the images. As -1≤ Q0≤1, a value of Q0 that is closer to 1 indicates better fusion performance.

RMSE =

 [I

ref

− I f ]2

(16)

M 1M 2

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 1. Artificially Distorted Aircraft Source Images and Fusion Results: (a) Aircraft 1,(b) Aircraft 2, (c) Aircraft 3, (d) Reference Image, (e) Fast-ICA Fusion Result, (f) PSO-ICA Fusion Result.

An Efficient Algorithm for Multi-focus Image Fusion Using PSO-ICA

165

Fig. 2. Fitness function value vs No. of Iterations Table 1. Performance Comparison of Fast-ICA and PSO-ICA fusion method Fusion Parameters Signal to Noise Ratio (SNR in DB) Image Quality Index (Qo) Root Mean Square Error (RMSE) Information Entropy (E) Signal to Noise Ratio (SNR in DB)

Fast-ICA

PSO-ICA

13.5384

17.6621

0.7881

0.9295

0.0968

0.1171

5.8284

6.0638

13.5384

17.6621

We trained the ICA bases with 8 x 8 non overlapping image patches from the input images. Then we implemented the fusion method using Fast-ICA and PSO ICA. We can see that we got better fusion performance using PSO-ICA both in visual quality and metric quality (PSNR, Q0). The reason may be, in Fast-ICA to find out the independent components no standard optimization method is used but we have introduced PSO in ICA to estimate the independent components. The parameters for PSO are taken as: Swarm size = 40, Dimension = size of W, w = 0.9 to 0.4, c1 = c2 = 2, Number of Iterations = 25.

6 Conclusion This paper has demonstrated that the proposed method outperforms the existing FastICA technique both visually and quantitatively. Though PSO and ICA are not new but their application to multi-focus image fusion is new. The problem is not multimodal, that’s why the standard PSO has been taken. The results are quite evident by the quantitative parameters as given in the table. Future work should include some more example figures to prove that proposed method is better. Also some more comparisons with other swarming methods can be included in future.

166

S. Agrawal, R. Panda, and L. Dora

References 1. De, I., Chanda, B.: A simple and efficient algorithm for multifocus image fusion using morphological wavelets. Signal Processing 86, 924–936 (2006) 2. Zaveri, T., Zaveri, M.: A Novel Two Step Region Based Multifocus Image Fusion Method. International Journal of Computer and Electrical Engineering 2(1) (February 2010) 3. van Leeuwen, J. (ed.): Computer Science Today. Recent Trends and Developments. LNCS, vol. 1000. Springer, Heidelberg (1995) 4. Seales, W., Dutta, S.: Everywhere-in-focus image fusion using controllable cameras. In: Proceedings of SPIE, vol. 2905, pp. 227–234 (1996) 5. Bloch, I.: Information combination operators for data fusion: a review with classification. IEEE Trans. SMC: Part A 26, 52–67 (1996) 6. Eltoukhy, H.A., Kavusi, S.: A computationally efficient algorithm for multi-focus image reconstruction. In: Proceedings of SPIE Electronic Imaging (June 2003) 7. Mukhopadhyay, S., Chanda, B.: Fusion of 2d gray scale images using multiscale morphology. Pattern Recognition 34, 1939–1949 (2001) 8. Yang, X., Yang, W., Pei, J.: Different focus points images fusion based on wavelet decompo-sition. In: Proceedings of Third International Conference on Information Fusion, vol. 1, pp. 3–8 (2000) 9. Zhang, Z., Blum, R.S.: Image fusion for a digital camera application. In: Conference Record of the Thirty-Second Asilomar Conference on Signals, Systems and Computers, vol. 1, pp. 603–607 (1998) 10. Maurer, D.E., Baker, J.P.: Fusing multimodal biometrics with quality estimates via a Bayesian belief network. Pattern Recognition 41, 821–832 (2008) 11. Hyvrinen, A., Karhunen, J., Oja, E.: Independent Component Analysis. Wiley, London (2001) 12. Nian, F., Li, W., Sun, X., Li, M.: An Improved Particle Swarm Op-timization Application to Independent Component Analysis. In: ICIECS 2009, pp. 1–4 (2009) 13. Mitianoudis, N., Stathaki, T.: Pixel-based and region-based image fusion using ICA bases. Information Fusion 8(2), 131–142 (2007) 14. Cvejic, N., Bull, D., Canagarajah, N.: Region-Based Multimodal Image Fusion using ICA Bases. IEEE Sensors Journal 7(5) (May 2007) 15. Nikolov, S., Bull, D., Canagarajah, N.: Wavelets for image fusion. In: Petrosian, A., Meyer, F. (eds.) Wavelets in Signal and Image Analysis, Computational Imaging and Vision Series, pp. 213–244. Kluwer Academic Publishers, Dordrecht (2001) 16. Kennedy, J., Eberhart, R.C.: Particle Swarm Optimization. In: Proceedings of IEEE International Conference on Neural Networks, Piscataway, NJ, pp. 1942–1948 (1995) 17. Liang, J.J., Qin, A.K., Suganthan, P.N., Baskar, S.: Comprehensive Learning Particle Swarm Optimizer for Global Optimization of Multimodal Functions. IEEE T. on Evolutionary Computation 10(3), 281–295 (2006) 18. Liang, J.J., Suganthan, P.N.: Dynamic Multi-Swarm Particle Swarm Optimizer. In: IEEE Swarm Intelligence Symposium, Pasadena, CA, USA, pp. 124–129 (June 2005)

Economic Emission OPF Using Hybrid GA-Particle Swarm Optimization J. Preetha Roselyn1, D. Devaraj2, and Subranshu Sekhar Dash3 1

SRM University, Kattankulathur-603203 [email protected] 2 Kalasalingam University, Srivilliputhur-626190 [email protected] 3 SRM University, Kattankulathur-603203 [email protected]

Abstract. This paper presents a Hybrid Genetic Algorithm (HGA) Particle Swarm Optimization (PSO) approach to solve Economic Emission Optimal Power Flow problem. The proposed approach optimizes two conflicting objective functions namely, fuel cost minimization and emission level minimization of polluted gases namely NOX, SOX and COx simultaneously while satisfying operational constraints. An improved PSO which permits the control variables to be represented in their natural form is proposed to solve this combinatorial optimization problem. In addition, the incorporation of genetic algorithm operators in PSO improves the effectiveness of the proposed algorithm. The validity and effectiveness have been tested with IEEE 30 bus system and the results show that the proposed algorithm is competent in solving Economic Emission OPF problem in comparison with other existing methods.

1 Introduction OPTIMAL Power Flow (OPF) is one of the most important function in modern energy management systems. The OPF problem aims to achieve an optimal solution to specific objective function such as fuel cost by adjusting the power system control variables while satisfying a set of operational and physical constraints. A number of mathematical programming based techniques have been proposed to solve the optimal power flow problem. These include gradient method [1-3], Newton method [4] linear programming [5,6] and Interior point method [7]. These traditional techniques rely on convexity to find the optimum solution. But due to the non-linear and non-convex nature of the OPF problem, the methods based on these assumptions do not guarantee to find the global optimum. Also, the discrete variables related to the tap changing transformer and shunt capacitors cannot be incorporated directly into the general optimal power flow programs. Recently Evolutionary Computation Techniques like Genetic Algorithm [8-9] and Evolutionary Programming [10] have been successfully applied to solve the OPF problems. After the Clean Air Act Amendments (Kyoto Protocol) in 1990, operating at minimum cost maintaining the security is no longer the sufficient criterion for dispatching electric power. Minimization of polluted gases such as NOx, SOx and COx in case of thermal generation power plants is also B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 167–175, 2011. © Springer-Verlag Berlin Heidelberg 2011

168

J.P. Roselyn, D. Devaraj, and S.S. Dash

becoming mandatory for the generation utilities in many countries. Hence, the minimization of pollution emission has to be considered along with the cost problem which makes OPF a multi-objective optimization problem [11-14]. Panigrahi et al [15] proposed a novel optimization approach for ELD problem using artificial immune system. The approach utilizes clonal selection principle and evolutionary approach wherein cloning of antibodies is performed followed by hypermutation. In the proposed approach the OPF problem is formulated as non linear constrained multi objective problem where fuel cost and environmental impact are treated as competing objectives. In addition, GA operators were included in PSO algorithm to improve the effectiveness of the algorithm. In this paper, continuous variables are represented as floating point numbers and discrete variables are represented as integers. Further for effectiveness, crossover and mutation operators which can directly operate on floating point numbers and integers are used. The effectiveness of the proposed GA -PSO has been demonstrated through IEEE 30 bus system.

2 Problem Statement List of Symbols Total fuel cost; FT P loss Network real power loss; Pi, Qi Real and Reactive Powers injected into network at bus i; Gij, Bij Mutual conductance and susceptance between bus i and bus j; Gii,Bii Self- conductance and susceptance of bus i; Pgi, Qg Real and Reactive power generation at bus i ; Reactive power generated by ith capacitor bank; QCi Tap setting of transformer at branch k ; tk Voltage magnitude at bus i; Vi Vj Voltage magnitude at bus j; θij

Sl gk NB NB-1 NPQ Ng Nc NT Nl

Voltage angle difference between bus i and bus j; Apparent power flow through the lth branch; Conductance of branch k; Total number of buses; Total Number of buses excluding slack bus; Number of PQ buses; Number of generator buses; Number of capacitor banks; Number of tap-Setting transformer branches; Number of branches in the system.

A. Fuel Cost Minimization In general, the optimal power flow problem is formulated as an optimization problem in which minimize a specific objective function is minimized while satisfying a

Economic Emission OPF Using Hybrid GA-Particle Swarm Optimization

169

number of equality and inequality constraints. Mathematically, this problem is stated as, Ng

Minimize FT =



Fi

(1)

i =1

2

Where Fi = a i PGi

+ bi PGi + ci $/hr

B. Emission Cost Minimization The problem formulation is same as that of real power dispatch problem, but emission coefficients in place of fuel coefficients are used and dispatching is done by allocation of power generation across various generation units [16]. The problem formulations for various emissions are given below: NG

Min

F ( x) =  α i + β i PGi + γ i PGi2

(2)

i =1

Where αi, ßi and γ i are coefficients of generator emission characteristics and vary for NOx, SOx and COx Gases. Subject to: (i) Load flow constraints: NB

Pi − Vi V j (Gij Cosθ ij + Bij Sinθ ij ) = 0, i = 1,2,.......N B−1

(3)

j =1

NB

Qi − Vi V j (Gij Sinθij − Bij Cosθij ) = 0, i = 1,2,......NPQ

(4)

j =1

(ii) Voltage constraint:

i ∈ NB

Vimin < Vi < Vimax

(5)

(iii) Generator reactive power generation limit:

i ∈ Ng

Qgimin < Qgi ≤Qgimax

(6)

(iv) Reactive power generation limit of capacitor banks QCi min < QCi ≤ QCi max

i ∈ NC

(7)

(v) Transformer tap setting limit: tkmin < tk < tkmax

k ∈ NT

(8)

(vi) Transmission line flow limit Sl < Slmax

l ∈ Nl

(9)

170

J.P. Roselyn, D. Devaraj, and S.S. Dash

The equality constraints given by Equations (3) and (9) are satisfied by running the power flow program. The active power generation (Pgi) (except the generator at the slack bus), generator terminal bus voltages (Vgi), transformer tap settings (tk) and reactive power generation of capacitor bank (Qci) are the optimization variables and they are self-restricted by the optimization algorithm.

3 Proposed Hybrid GA Particle Swarm Optimization Particle Swarm Optimization (PSO) is a simple and efficient population based optimization method proposed by Kennedy and Eberhart [17]. Let x i and v i denote the positions and the corresponding flight speed (velocity) of the particle i in a continuous search space, respectively. The modified velocity and position of each particle can be calculated using the current velocity and the distance from the pbest and gbest as follows: Vik+1= wVik +c1 rand1(.)(pbesti-sik) + c2 rand2(.) .(gbest-sik)

(10)

(11) Where: vik: velocity of agent i at iteration k, w: weighting function, cj : weighting factor, rand : uniformly distributed random number between 0 and 1 sik : current position of agent i at iteration k, pbesti : pbest of agent i, gbest: gbest of the group. vit+1 : velocity of agent i at iteration t+1 xit : position of agent i at iteration t xit+1 : position of agent i at iteration t+1 In the velocity updating process, the value of the parameters such as w, c1, c2 and k should be determined in advance. In the proposed GA-PSO for OPF problem, genetic operators were incorporated in the PSO algorithm which improves the effectiveness of the proposed algorithm. In the proposed algorithm, before population is updated, elitism was executed and best solutions from previous population were retained, while velocity and particle position were updated based on a component from GA algorithm, i.e. Mutation which is incorporated for updating particle position and generating new population. This was done because as PSO algorithm progresses, and convergence towards global best solution is attained, process of updating population is slowly halted, mutation according to fitness keeps on updating population in order to further optimize solution and converge towards even better solution.

Economic Emission OPF Using Hybrid GA-Particle Swarm Optimization

4

171

PSO Implementation

While applying PSO for the OPF problem, the following issues need to be addressed: • • 4.1

Problem Representation and Fitness evaluation.

Problem Representation

Implementation of PSO for a problem starts with the parameter encoding. Each individual in the population represents a candidate solution. With mixed form of representation, an individual in the PSO population for the OPF problem will look like the following: 97.5 Pg2

100.8 ... 250.70 Pg3

0.981

Pgn

Vg1

0.970 … 1.05 Vg2

3 ..

Vgn

t1

1 ...

8

t2

tn

2 Qc1

1 ….. 8 Qc2

Qcn

This coding scheme avoids the redundant value mapping or the introduction of artificial constraints which is necessary if binary coding is used to represent the transformer tap setting and the reactive power generation of capacitor. Also, with direct representation of the solution variables, the computer space required to store the population is reduced. 4.2 Evaluation Function PSO searches for the optimal solution by maximizing a given fitness function, and therefore an evaluation function which provides a measure of the quality of the problem solution must be provided. In the OPF problem under consideration, the objective is to minimize the total fuel cost satisfying the constraints. With the inclusion of the penalty function the new objective function becomes, N max 2 ) +K Miin f = F + K (P − P T s sl sl v

PQ



i =1

N lim 2 ( V −V ) + K i i q

g

N l



lim 2 )

lim )2 + K ( S −S  ( Qgi − Qgi l l l l =1 i =1

(12)

Where Ks, Kv, Kq, Kl and Ks are the penalty factors for the slack bus power output, bus voltage limit violation, generator reactive power limit violation, line flow violation and voltage stability limit violation respectively. The success of the approach lies in the proper choice of these penalty parameters. Since PSO maximizes the fitness function, the minimization objective function f is transformed to a fitness function to be maximized as, Fitness = k/f where k is a large constant.

172

J.P. Roselyn, D. Devaraj, and S.S. Dash

5 Simulation Results The proposed GA based PSO approach has been applied to solve the economic emission optimal power flow problem in I EEE-30 bus system. The buses 10, 12, 15, 17, 20, 21, 23, 24 and 29 are identified for reactive power injection. The generator cost coefficients and the t r ansmission line parameters are taken from [2]. The emission characteristics of Sox, NOx and COx are given in Appendix A1. The OPF algorithm solved using GA based PSO was implemented using the MATLAB program and was executed on Intel Core Duo 1.66GHz, with a 2 GB memory computer. Two different cases were considered for simulation, one minimizing fuel cost without considering the emission function and the next one to solve the OPF problem by including the emission function in the problem formulation. The results of these simulations are presented below. Case 1: Minimizing Generation Cost (Single Objective Optimization) In this case, PSO algorithm was applied to solve the optimal power flow with single objective function to minimize fuel cost by identifying the optimal control parameters using different PSO parameter settings. The minimum fuel cost obtained using PSO based OPF is 802.06 $/hr with no limit violations. In addition different combinations of GA operators were included in the OPF problem and are mentioned in Table 1. From this table, it is clear that crossover and mutation operators when incorporated into PSO algorithm provides better results than the other combinations. Table 1. Different combinations of genetic operators in PSO algorithm

GA operators

Pm

Mutation Mutation, elitism, selection crossover Elitism & mutation Elitism, selection & mutation Elitism, crossover & mutation Crossover &mutation

&

Pc

0.2

0

Minimum fuel cost ($/hr) 802.31

0.001

0.9

803.4

0.001 0.0013 0.05 0.001

1 0.6

802.02 802.09 802.35 801.81

The optimal settings corresponding to the optimal combination of GA based PSO is given in Table 2. Further, the minimum cost solution obtained by the proposed approach is less than the values reported in literature for the IEEE 30-bus system and is shown in Table 3. This shows the powerfulness of the proposed algorithm to obtain the optimal solution in the OPF problem.

Economic Emission OPF Using Hybrid GA-Particle Swarm Optimization

173

Table 2. Optimal control settings for GA-PSO for OPF problem Control Variables

Variable Setting

Control Variables

Variable setting

P1

176.37

T11

3.0

P2

48.88

T12

0.0

P5

21.47

T15

2.0

P8

21.98

T36

0.0

P11

12.03

Qc10

5

P13

12.00

Qc12

5

V1

1.0499

Qc15

5

V2

1.0366

Qc17

5

V5

1.0096

Qc20

4

V8

1.0171

Qc21

5

V11

1.0999

Qc23

3

V13

1.1000

Qc24

5

Qc29

1

Cost

801.81 $/hr

Table 3. Comparison of fuel cost

Algorithm Tabu Search [18] Improved GA [14] Modified DE [19] PSO Hybrid PSO

Fuel cost ($/hr) 802.50 802.19 802.37 802.06 801.81

Table 4. Comparison of fuel cost and emission level with PSO and GA-PSO

Gas Optimization technique Fuel cost ($/hr) Emission (Kg/hr)

Nox GA based PSO 849.511 848.467 PSO

1263.7

1263.3

Sox GA based PSO 855.546 847.710

PSO

2892.3

2884.7

Cox GA based PSO 872.0426 862.579

PSO

15043

15008

174

J.P. Roselyn, D. Devaraj, and S.S. Dash

Case 2: Hybrid GA-PSO with minimizing Fuel Cost and Emission functions In the proposed multi-objective problem, the problem was handled as a multi objective optimization problem with fuel cost and controlled emission as objectives to be minimized simultaneously. The fuel cost objective problem is combined with three combinations of polluted gases namely NOx, SOx and COx and the simulation results are given in Table 4. From this table, it is clear that multi objective GA-PSO provides better results than multi-objective PSO for all combinations of objective functions.

6 Conclusion This paper has proposed an improved Particle Swarm Optimization incorporating genetic algorithm operators to solve the economic emission OPF problem. The performance of PSO with different combination of genetic operators has been explored. Simulation results on the IEEE 30-bus system are presented to illustrate the effectiveness of the proposed approach to solve the economic emission OPF problem. Hence by incorporating emission control into OPF problem, the proposed method converges fast, provides better outputs and also occupies less memory space. In future, MOPSO discussed in [20] will be considered to solve the multi-objective OPF problem.

References 1. Dommel, H.W., Tinney, W.F.: Optimal power flow solutions. IEEE Transactions on Power Apparatus and Systems PAS-87 (10), 1866–1876 (1968) 2. Alsac, O., Scott, B.: Optimal load flow with steady state security. IEEE Transactions on Power Systems PAS 93(3), 745–751 (1974) 3. Lee, K.Y., Park, Y.M., Ortiz, J.L.: Optimal real and reactive power dispatch. Electric Power Systems Research 7, 201–212 (1984) 4. Sun, D.I., et al.: Optimal power flow by Newton approach. IEEE Transactions on PAS 103(10), 2864–2880 (1984) 5. Stott, B., Hobson, E.: Power system security control calculations using linear Programming. IEEE Transactions on Power Apparatus and Systems, PAS 97, 1713–1931 (1978) 6. Mangoli, M.K., Lee, K.Y.: Optimal real and reactive power control using linear Programming. Electric Power Systems Research 26, 1–10 (1993) 7. Momoh, J.A., Zhu, J.Z.: Improved interior point method for OPF problems. IEEE Transactions on Power Systems 14, 1114–1120 (1999) 8. Lai, L.L., Ma, J.T., Yokoyama, R., Zhao, M.: Improved genetic algorithm for optimal power flow under both normal and contingent operation states. Electrical Power and Energy Systems 9(5), 287–292 (1997) 9. Bauirtzis, A.G., et al.: Optimal power flow by Enhanced genetic algorithm. IEEE Transactions on Power Systems 14, 1114–1120 (1999) 10. Yuryevich, J., Wang, K.P.: Evolutionary Programming based optimal power flow algorithms. IEEE Transactions on Power Systems 14(4), 1245–1250 11. Deb, K.: Multi-objective evolutionary algorithm. John Wiley & Sons publications 12. Abido, M.A.: Multi-objective particle swarm optimization for environmental/economic dispatch problem. Electric Power System Research 79, 1105–1113 (2009)

Economic Emission OPF Using Hybrid GA-Particle Swarm Optimization

175

13. Wang, L., Singh, C.: Stochastic combined heat and power dispatch based on multiobjective particle swarm optimization. International Journal of Electrical Power and Energy Systems 30, 226–234 (2008) 14. Devaraj, D., Preetha Roselyn, J.: Improved genetic algorithm for voltage security constrained optimal power flow problem. Int. Journal Energy Technology and Policy 5(4), 475–488 15. Panigrahi, B.K., Yadav, S.R., Agrawal, S., Tiwari, M.K.: Aclonal algorithm to solve economic load dispatch. Electric Power System Research 77, 1381–1389 (2007) 16. Bouktir, T., Labdani, R., Slimani, L.: Economic Power Dispatch of Power System With Pollution Control Using Multi objective Particle Swarm Optimization. Journal of Pure and Applied Sciences 4, 57–77 (2007) 17. Kennedy, J., Eberhart, R.: Swarm intelligence. Morgan Kaufmann publishers (2001) 18. Ongsakul, W., Tantimaporn, T.: Optimal power flow by improved evolutionary Programming. International Journal of Electric power Component and Systems (34), 79– 95 (2006) 19. Sayah, S., Zehar, K.: Modified differential evolution algorithm for optimal power flow with non-smooth cost functions. Energy Conversion and Management 49, 3036–3042 (2008) 20. Zhou, A., Ru, B.-Y., Huiti, Zhao, S.-Z., Suganthan, P.N., Zhang, Q.: Multi-objective Evolutionary Algorithm:A survey of the state of the art. Swarm and Evolutionary Computation 1, 32–49 (2011)

Appendix A1. Emission cost co-efficients

Application of Improved PSO Technique for Short Term Hydrothermal Generation Scheduling of Power System S. Padmini1, C. Christober Asir Rajan2, and Pallavi Murthy3 1

2

Department of Electrical and Electronics Engineering, SRM University, Chennai, India Department of Electrical and Electronics Engineering Pondichery University, Chennai, India 3 Department of Electrical and Electronics Engineering, SRM University, Chennai, India [email protected], [email protected], [email protected]

Abstract. This paper addresses short-term scheduling of hydrothermal systems by using Particle Swarm Optimization (PSO) algorithm. Particle Swarm Optimization is applied to determine the optimal hourly schedule of power generation in a hydrothermal power system. The developed algorithm is illustrated for a test system consisting of one hydro and one thermal plant respectively. The effectiveness and stochastic nature of proposed algorithm has been tested with standard test case and the results have been compared with earlier works. It is found that convergence characteristic is excellent and the results obtained by the proposed method are superior in terms of fuel cost . Keywords: Hydrothermal Scheduling, Particle Swarm Optimization.

1

Introduction

Now a days, because of the increasing competition in power market, scheduling the Hydro and Thermal energy in the most economic manner has become an important task in modern power systems. The main objective of hydrothermal operation is to minimize the total system operating cost, represented by the fuel cost for the systems thermal generation subject to the operating constraints of hydro and thermal plant over the optimization interval. In short range problem the water inflows is considered fully known and is constrained by the amount of water available for draw down in the interval. The short term hydro thermal scheduling problems have been solved by various methods. These methods which have been reported in the literature includes classical methods such as Langrage Multiplier Gradient Search and Dynamic programming stochastic search algorithm such as simulated annealing (SA) [6], Genetic algorithm (GA) [3], Evolutionary Programming (EP) [7,8] and Particle Swarm Optimization (PSO) [2,9,10,11]. The PSO technique can generate high-quality solution within shorter calculation time and more stable convergence characteristic than other stochastic methods. Many researches are still for proving its potential in solving complex power system problems. B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 176–182, 2011. © Springer-Verlag Berlin Heidelberg 2011

Application of Improved PSO Technique

2

Problem Formulation

2.1

Objective Function

177

The main objective of Hydro Thermal Scheduling is minimizing the thermal generation cost by satisfying the hydro and thermal constraints. Hydro thermal scheduling is the optimization of a problem with non-linear objective function, the objective function to be minimized can be written as: The problem formulation is same as that of real power dispatch problem, but emission coefficients in place of fuel coefficients are used and dispatching is done by allocation of power generation across various generation units [16]. The problem formulations for various emissions are given below:

min( f ) =  n j f j (PT j ) N

(1)

j =1

Where j = 1, 2, 3 ………N schedule interval f j PT j is the thermal generation cost in $/h

(

)

nj is the number of hours in jth interval Subject to:

(P

Power balance constraint

dj

)

+ PLoss j − (PH j + PT j ) = 0

The transmission loss is given by

PLoss = k (PH j )

(2) (3)

The hydro generation is consider to be a function of discharge rate only

q j = g (PH j )

(4)

q max > q j > q min

(5)

Discharge rate limits Thermal generation limits

PTmax > PT j > PTmin

(6)

PH max > PH j > PH min

(7)

Hydro generation limits Reservoir volume

Volume j +1 = Volume j + n j (r j − q j − s j )

Where n j is the number of hours in jth interval rj is the water inflow rate in jth interval q j is the water discharge rate in jth interval s j is the water spillage rate in jth interval

(8)

178

S. Padmini, C. Christober Asir Rajan, and P. Murthy

Reservoir storage limits

Volumemax > Volume j > Volumemin

3

(9)

Particle Swarm Optimization

Particle swarm optimization (PSO), first introduced by Kennedy and Eberhart in 1995, is one of the modern heuristic optimization algorithms. Among all the stochastic search algorithms PSO gives the reasonable solution in a short CPU time. The application of PSO technique into nonlinear discontinuous constrained power system problems such as Economic Dispatch (ED), Unit Commitment (UC), state Estimation (SE), Load Forecasting (LF), Optimal Power Flow (OPF), etc have been reported by several authors. This method has been developed under the scope of artificial life where PSO is inspired by the natural phenomenon of fish schooling or bird flocking. PSO is initialized with a group of random particles and then searches for optima by updating generation. In every iteration each particle is updated by two best values. First one is based on its own best exploration called pbest. Another best value is based on best swarm overall experience called gbest. After finding two best values the particle velocity and positions are updated by following equations.

(

)

(

V jk +1 = WV jkk + c1 rand1 ( ) pbest − x kj + c 2 rand 2 ( ) gbest − x kj

x kj +1 = x kj + K * V jik +1

)

(10) (11)

Where

V jk = Velocity of individual particle (i) at iteration k W = Weight parameter

W = Wmax −

Wmax − Wmin * iter itermax

(12)

c1 & c2 : Acceleration Constant c1 = 2.5= c2 rand1( ) & rand2( ) : Uniform random number in range [0,1] pbest : Previous best value gbest : Global best value K : Constriction factor

4

Proposed PSO Algorithm for Hydrothermal Scheduling

The various possible particles are Hydro generation, thermal generation, water discharge rate and reservoir volume for hydro thermal scheduling problem. One can select any of the above as particle. The work shows that selecting discharge as a particle gives the best result. The proposed PSO based HTS algorithm with reservoir volume as particle is presented here.

Application of Improved PSO Technique

179

Step 1: Input parameters of the system and specify the upper and lower boundaries of each variable. Step 2: Initialize randomly the particles of the population. Step 3: Let qj be the dishcarge rate denoting the particles of population to be evolved.It is the discharges of turbines of reservoirs at various intervals.Then knowing the hydro discharges, storage volumes of reservoirs Vj are calculated using Eq.(8).Then PH and PT is calculated for all the intervals. Step 4: Compare each particle with its Pbest.The best evaluation value among the Pbest is denoted as gbest. Step 5: Update the iteration count and update the velocity and particle position uisng Eq. (10) and Eq. (11)respectively. Step 6: Each particle is evaluated according to its updated position, only when satisfied by all constraints. If the evaluation value of each particle is better than the previous Pbest. The current value is set to be Pbest. If the best Pbest is better than gbest, the value is set to be gbest. Step 7: If the stopping criterion is reached, then print the result and stop; otherwise repeat steps 2–6.

5

Test System and Results

In this study, a test system consisting of one hydro unit and one thermal unit respectively. The entire optimization period is three days and it has been divided into 6 intervals and each interval is of 12 hours. Three days’ twelve hour based load curve is given in Table 1. Table 1. Load Demand

Interval

Day

Time Interval

Demand (MW)

1

1

0-12h

1200

2

1

12-24h

1500

3

2

0-12h

1100

4

2

12-24h

1800

5

3

0-12h

950

6

3

12-24h

1300

180

S. Padmini, C. Christober Asir Rajan, and P. Murthy

Water discharge rate qj qj

=330 + 4.97* PHj

if

0 < PHj < 1000

=5300 + 12 (PHj - 1000) + 0.05 (PHj - 1000) if

PHj > 1000

qj = 0 Volume start

if =

PHj = 0 100000 acre – ft

Volume End

=

60000

Volume mix

=

120000 acre – ft

Volume min

=

60000

acre – ft /h

2

(13) (14)

acre – ft /h

acre – ft

acre – ft

The objective function, hydro constraints and thermal constraints are given below. The cost characteristic of thermal plant is given by: f (PTj) = 0.000184 PHj2 + 9.2 PHj + 575 $ / h 150 < PTj < 1500 0 < PTj < 1100

(15)

MW MW

The reservoir is considered to have a constant water inflow (rj) of 2000 acre-ft/h. The spillage rate j s is considered to be zero. Transmission loss is taken to be negligibly small. In applying the proposed PSO algorithm for the test system, the appropriate values of population size NP and maximum iteration number Nmax are set to the values of 30 and 100 , respectively.Table-2 gives the results of PSO by considering water discharge as a particle.

PH

Volume

Discharge

(MW)

(MW)

(acre ft)

Rate (acreft/hr)

1

812.54

387.45

96931.91

2255.674

2

801.58

698.41

75318.31

3801.133

3

1100

0

99314.31

0

4

804.72

995.27

59996.04

5276.52

5

950

0

83996.04

0

6

561.56

738.43

59996.04

4000

Cost ($)

PT

606423.16

Interval

Table 2. Results of Proposed Method

Application of Improved PSO Technique

181

Table 3. Comparison of Results

S.No

Author

Year

Method

Cost

1

Wood.A.J

1984

GS

709877.38

2

Sinha.et.al

2003

IFEP

709862.05

3

Wong.K.P.et.al

1994

SA

709874.36

4

Sinha.et.al

2006

GAF

709863.70

5

Sinha.et.al

2006

CEP

709862.65

6

Sinha.et.al

2006

FEP

709864.59

7

Sinha.et.al

2006

PSO

709862.04

8

D.S.Suman.et.al

2006

Hybrid EP

703180.26

9

Proposed

2011

PSO

606423.16

The result of the proposed algorithm has been compared with the results of earlier works in Table-3. The proposed algorithm was more successful in reducing the cost searching the entire search space than all other earlier methods. All the earlier work results have taken from references [1, 3, 6, 7, 8 and 9]. Table-3 gives the comparison of earlier works and proposed method. The convergence characteristics of the proposed method is given in figure-1.

182

S. Padmini, C. Christober Asir Rajan, and P. Murthy

6 Conclusion In this paper, a particle swarm optimization (PSO) based algortihm has been proposed for solving Short term hydrothermal scheduling problem by taking the discharge rate as particle.The results shows that best optimal solutions can be obtained by particle swarm optimization method,when compared with already existing techniques.A variant of particle swarm optimizer can be used whereby all other particles’historical best information can be used to update the particle’s velocity.This strategy enables the diversity of the swarm to be preserved to discourage premature convergence.This can be used as the future work to yield better convergence .

References 1. Wood, A.J., Wollenberg, B.F.: Power Generation, Operation and Control. Operation and Control. John Wiley and Sons, New York (1984) 2. Umayal, S.P., Kamaraj, N.: Stochastic Multi Objective Short term Hydrothermal Scheduling Using Particle Swarm Optimization (2005) 3. Ferrero, R.W., Rivera, J.F., Shahidehpour, S.M.: A dynamicprogramming two-stage algorithm for long-termhydrothermal scheduling of multireservoir systems. IEEE Transactions on Power Systems 13(4), 1534–1540 (1998) 4. Chang, W.: Optimal Scheduling of Hydrothermal System Based on Improved Particle Swarm Optimization 5. Samudi, C., Das, G.P., Ojha, P.C., Sreeni, T.S., Cherian, S.: Hydro thernal Scheduling using Particle Swarm Optimization (2008) 6. Wong, K.P., Wong, Y.W.: Short-term hydrothermal scheduling, part-I: Simulated. annealing approach. IEE Proc., Part- C 141(5), 497–501 (1994) 7. Sinha, N., Chakrabarti, R.: Fast Evolutionary Programming Techniques For Short-Term Hydrothermal Scheduling. IEEE Trans. PWRS 18(1), 214–219 (2003) 8. Suman, D.S., Nallasivan, C., Henry, J., Ravichandran, S.: A Novel Approach for ShortTerm Hydrothermal Scheduling Using Hybrid Technique. In: IEEE Power India Conference, April 10-12 (2006) 9. Sinha, N., Lai, L.-L.: Meta Heuristic Search Algorithms for Short-Term Hydrothermal Scheduling. In: International Conference on Machine Learning and Cybernetics, Dalian (2006) 10. Hotaa, P.K., Barisala, A.K., Chakrabarti, R.: An improved PSO technique for short-term optimal hydrothermal scheduling. Electric Power Systems Research 79(7), 1047–1053 (2009) 11. Liang, J.J., Qin, A.K., Suganthan, P.N., Baskar, S.: Comprehensive Learning Particle Swarm Optimizer for Global Optimization of Multimodal Functions. IEEE T. on Evolutionary Computation 10(3), 281–295 (2006)

Multi-objective Workflow Grid Scheduling Based on Discrete Particle Swarm Optimization Ritu Garg and Awadhesh Kumar Singh Computer Engineering Department, National Institute of Technology, Kurukshetra, Haryana, India [email protected], [email protected]

Abstract. Grid computing infrastructure emerged as a next generation of high performance computing by providing availability of vast heterogenous resources. In the dynamic envirnment of grid, a schedling decision is still challenging. In this paper, we present efficient scheduling scheme for workflow grid based on discrete particle swarm optimization. We attempt to create an optimized schedule by considering two conflicting objectives, namely the execution time (makespan) and total cost, for workflow execution. Multiple solutions have been produced using non dominated sort particle swarm optimization (NSPSO) [13]. Moreover, the selection of a solution out of multiple solutions has been left to the user. The effectiveness of the used algorithm is demostrated by comparing it with well known genetic algorithm NSGA-II. Simulation analysis manifests that NSPSO is able to find set of optimal solutions with better convergence and uniform diversity in small computation overhead.

1

Introduction

With the rapid development of networking technology, grid computing [4] has emerged as a promising distributed computing paradigm that enables large-scale resource sharing and collaboration. One of the key challenges of heterogeneous systems is the scheduling problem. Scheduling of computational tasks on the Grid is a complex optimization problem, which may require consideration of several scheduling criteria. Usually, the most important criterion is the task execution time, cost of running a task on a machine, reliability, resource utilization etc. The optimization of scheduling problem is NP-complete, so numerous heuristic algorithms have been proposed in literature [3]. Many heuristics have also been proposed for workflow scheduling in order to optimize a single objective [9], [10], [16], [20]. To achieve better solution quality, many meta-heuristic methods have been presented for job scheduling such as simulated annealing (SA) [2], genetic algorithm (GA) [15], ant colony optimization (ACO) [14] and tabu search[6]. Defining the multiple objectives for the task scheduling problem for generating efficient schedules at reduced computational times are of research interest in the recent days. For multi objective optimization to independent task scheduling [1] used the fuzzy particle swarm optimization and [11] used the discrete particle swarm optimization. These methods combine the multiple objectives into scalar cost function B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 183–190, 2011. © Springer-Verlag Berlin Heidelberg 2011

184

R. Garg and A.K. Singh

using the weight factors, which convert the problem into single objective problem prior to optimization. Generally, it is very difficult to accurately select these weights as small perturbations in weights leads to different solutions. Hence, in this paper we introduced the multi objective optimization approach based on discrete particle swarm optimization to generate Pareto optimal solutions, which is the set consisting of all non-dominated solutions. A solution is called non-dominated solution if it is best at least in one objective with respect to others. Pareto optimal solutions are preferred to single solution in real life applications. In this paper, we considers the two objectives for task scheduling keeping in view the tradeoff between two conflicting objectives of minimizing the makespan and total cost under the specified deadline and budget constraint. Towards the goal of obtaining Pareto optimal set, we applied non dominated sort PSO (NSPSO) [13] as it performs better against a set of well known test functions that are presumed difficult. Rest of the paper is organized as follows. Section 2 specifies the problem definition. In section 3, describes the formulation of discrete particle swarm optimization for multi objective workflow grid scheduling. In section 4, we explained the procedure of non dominated sort particle swarm optimization algorithm which is used to obtain Pareto optimal solutions. Section 5 discusses the simulation analysis and finally section 6 gives the conclusion.

2

Problem Definition: Workflow Grid Scheduling

We define workflow Grid scheduling as the problem of assigning different precedence constraint tasks in the workflow to different available grid resources. We model the application as a task graph: Let G = (V, E) be a directed acyclic graph (DAG), with V as the set of n tasks ti є V, 1 ≤ i ≤ n and E is the set of edges representing precedence constraint among the tasks eij = (ti, tj) ∈ E, 1 ≤ i ≤ n, 1 ≤ j ≤ n, i ≠ j. Associated to each edge is the amount of data required to send from task ti to tj if they are not executed on the same resource. Let set R represent the m number of resources which are available in the grid and resource rj є R is associated with two values: Time and cost of executing the task on resource rj. Every task ti has to be processed on resource rj until completion. In our work, scheduling solution is represented as the task assignment string corresponding to the scheduling order string. Task assignment string is the allocation of each task to the available time slot of the resource capable of executing the task, and the scheduling order string encodes the order to schedule tasks. The ordering of tasks in the scheduling order string must satisfy the task dependencies. The execution optimization problem is to generate task assignment string S, which maps every ti onto a suitable rj to achieve the multi objective below: Minimize Time(S) = maxtime (ti) where ti є V and 1≤i ≤n

(1)

where ti є V and 1≤i ≤n

(2)

Minimize Cost(S) = ∑ cost (ti)

Subject to Cost(S) < B and Time(S) < D Where B is the cost constraint (Budget) and D is the time constraint (Deadline) required by users for workflow execution.

Multi-objective Workflow Grid Scheduling

3

185

Discrete Particle Swarm Optimization for Workflow Grid Scheduling

In this paper, we used the version of discrete particle swarm optimization (DPSO) [11] to solve the problem of workflow grid scheduling. PSO is a self adaptive global search optimization technique introduced by Kennedy and Eberhart [12] and it relies on the social behavior of the particles. In every generation, each particle adjusts its trajectory based on its best position (local best) and the position of the best particle (Global best) of the entire population. One of the key issues in designing successful PSO algorithm is the representation step, i.e. finding a suitable mapping between problem solution and PSO particle. For optimization of workflow grid scheduling problem, solution is represented as task assignment string (S) as mentioned in section 2. To represent S, we setup an n dimension search space corresponding to n number of tasks and each dimension represents the discrete value corresponding to m number of resources. Here, solutions or task assignment strings are encoded as m×n matrix, called position matrix where m is the number of available resources and n is the number of tasks. Let Xk is the position matrix of kth particle then Xk (i,j)

∈ {0,1} (∀ i,j), i ∈ {1,2,...m}, j ∈ {1,2,..., n}.

(3)

where Xk (i, j) = 1 means that jth task is performed by ith resource. Hence, in each column of the matrix only single element is 1 and others are 0. For example, Fig.1. shows the mapping between one possible task assignment strings to the particle position matrix in PSO domain. Task Assignment String {[T1 : R3], [T2 : R1], [T3 : R3], [T4 : R2], [T5 : R1], [T6 : R3]} Particle Position Matrix

T1

T2

T3

T4

T5

T6

R1

0

1

0

0

1

0

R2

0

0

0

1

0

0

R3

1

0

1

0

0

1

Fig. 1. Mapping of Task assignment string to Particle Position matrix

Velocity of each particle is again an m×n matrix whose elements are in range [-Vmax, Vmax]. If Vk is the velocity matrix of kth particle, then: Vk (i,j) ∈ [-Vmax, Vmax ], (∀ i, j), i ∈ {1,2,...m}, j ∈ {1,2,..., n}.

(4)

Also, Pbest and Gbest are m×n matrices and their elements assume value 0 or 1 as in the case of position matrices. Pbestk represents the best position that kth particle has

186

R. Garg and A.K. Singh

visited since the initial time step and Gbestk represents the best position that kth particle and its neighbors have visited since the algorithm was initiated. For updating Pbestk and Gbestk in each time stamp we are using the non dominated sort multi objective PSO algorithm as mentioned by procedure of NSPSO in section 4. For particle updating, we are first updating velocity matrix according to (5) and then finally position matrix is updated using (6). V

(i,j) = ω. V

X

i, j

(i,j) +c1r1 (Pbest

1,

V

(i,j) - X

i, j

(i,j)) + c2r2 (Gbest

max V

i, j

i

(i,j) - X

1,2, . . .

(i,j))

(5)

(6)

0,

Using equation (6), each column of position matrix, the value 1 is assigned to the element whose corresponding element in velocity matrix has maximum value in its corresponding column. If in a column of velocity matrix there are more than one element with max value, then one of these elements is selected randomly and 1 is assigned to its corresponding element in the position matrix. A particle represented as position matrix Xk is formulated from task assignment string (S). Initially, S representing resource on which a task will execute is defined randomly. The fitness functions Ftime(S) and Fcost(S) are formed in order to evaluate individuals according to makespan and cost of the schedule respectively. These fitness functions are calculated from Equation (1) and (2) by adding the penalty. On the violation of deadline and budget constraints, penalty is added respective to objective function, otherwise not.

4

Multi Objective Optimization Algorithm Used

To optimize workflow grid scheduling under two conflicting objective of makespan and total cost, we are using non dominated sort particle swarm optimization (NSPSO) [13] approach. NSPSO extends the basic form of PSO by making a better use of particle’s personal bests and offspring for effective non-domination comparisons. The steps of basic NSPSO procedure are as follows: NSPSO Procedure 1. Create and initialize m×n dimensional swarm with N particles randomly. The initial velocity for each particle is also initialized randomly but in the range [-Vmax, Vmax]. The personal best position of each particle (Pbestk), is set to Xk. 2. Evaluate each particle in the swarm. 3. Apply non-dominated sorting on the particles. 4. Calculate crowding distance of each particle. 5. Sort the solutions based on decreasing order of crowding distance.

Multi-objective Workflow Grid Scheduling

187

6. Select randomly Gbest for each particle from a specified top part (e.g. top 5%) of the first front F1; for all particles 7. Calculate the new velocity V based on Equation (5) and new position X from and Pbest . Equation (6) using the determined Gbest 8. Create a new population of size 2N by combining the new position and their personal best, X ∪ Pbest . 9. Apply non-dominated sorting on 2N particles and calculate the crowding distance for each particle. 10. Generate a new set of N solutions by selecting solutions from non-dominated fronts F1, F2 and so on using the crowding distance. The N solutions form the personal best for the next iteration. 11. Go to step 2 till the termination criteria is met.

5

Simulation Results and Discussion

We used GridSim [5] toolkit in our experiment to simulate the scheduling of workflow tasks. GridSim is a java based toolkit for modeling and simulation of resource and application scheduling in large-scale parallel and distributed computing environment such as Grid. In our test environment, we simulated the balanced workflow consisting of 20 tasks on 8 virtual resources and these resources are maintained by different organizations in the grid. Links between resources are established through a router so that direct communication can take place between resources. Computational rating (Million instructions per second) and computational cost (in dollars) of each resource is generated with non-uniform distribution. Number of data units required by one task from another task in the workflow is also generated non-uniformly. In order to generate valid schedule which can meet both deadline and budget constraints specified by the user, two algorithms HEFT [10] and Greedy Cost were used to make deadline and budget effectively. HEFT is a time optimization scheduling algorithm in which workflow tasks are scheduled on minimum execution time heterogeneous resources irrespective of utility cost of resources. So HEFT gives minimum makespan (Timemin) and maximum total cost (Costmax) of the workflow schedule. Greedy Cost is a cost optimization scheduling algorithm in which workflow tasks are scheduled on cheapest heterogeneous resources irrespective of the task execution time. Thus Greedy Cost gives maximum makespan (Timemax) and minimum total cost (Costmin) of the workflow schedule Thus Deadline (D) and Budget (B) are specified as: D = Timemax – 0.1(Timemax - Timemin)

(7)

B = Costmax – 0.1(Costmax - Costmin)

(8)

To measure the effectiveness and validity of NSPSO algorithm for workflow grid scheduling problem, we have implemented a probabilistic GA based technique known

188

R. Garg and A.K. Singh

as non dominated sort genetic algorithm (NSGA-II) [7]. To implement the NSGA-II we have taken binary tournament selection, two point crossover and replacing mutation. Each experiment was repeated 20 times with different random population initialization. Initial population was seeded with two solutions obtained by heuristics namely LOSS-II and modified GAIN-II. LOSS-II and modified GAIN-II were used to generate a solution with minimum makespan (total time) while meeting budget constraint and minimum total cost while meeting deadline constraint respectively. This is done because population with seeding which contains two already optimized boundary solutions gives good and fast convergence with better spread rather than generating all solutions randomly [20]. In order to compare the performance of algorithms, we have run the algorithms over 200 generations with the initial population size of 50. The Pareto optimal solutions obtained with NSPSO and NSGAII for bi-objective workflow grid scheduling problem are shown in Fig. 2. From a typical run shown in Fig. 2 we can see that most of the solutions obtained with NSPSO are lie on the better front as compared to NSGA-II while preserving uniform diversity between solutions.

Cost/Budget

1 NSGA-II

0.9

NSPSO

0.8 0.7 0.6 0.5 0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Makespan/Deadline Fig. 2. Obtained Pareto Optimal Solutions with NSPSO and NSGA-II

5.1

Performance Evaluation: GD, Spacing

For the performance comparison between NSPSO and NSGA-II we conducted our experiment over 20 runs and then average of these runs has been taken for evaluation. To measure the quality of evolutionary algorithms, we used two metrics Generational Distance (GD) and Spacing [8]. GD is the well known convergence metric to evaluate the quality of an algorithm against the reference front P*. The reference front P* was obtained by merging solutions of algorithms considered. On the other side, Spacing

Multi-objective Workflow Grid Scheduling

189

metric was used to evaluate diversity among solutions. The results obtained with these metrics for each algorithm are depicted in Table 1. The average value of the GD and spacing metric corresponding to NSPSO is less as compared to other algorithm considered i.e, NSGA-II. The result confirms the better convergence towards the real Pareto optimal front. Further, the low value of standard deviation shows that algorithm converges almost in every execution. Table 1. GD, Spacing Metric Results for the algorithms used

GD Metric Spacing Metric

6

NSGA-II

NSPSO

Avg

0.0231861

0.021173

Std. Dev.

0.001567

0.001480

Avg

0.042911

0.040233

Std. Dev.

0.003558

0.003324

Conclusion and Future Work

The current work emphasizes on the planning and optimizing the workflow scheduling in the grid. In this paper, we have used a version of discrete particle swarm optimization to represent the scheduling of workflow grid. The multi-objective non dominated sort particle swarm optimization (NSPSO) approach has been used to find solutions in the entire Pareto optimal front in order to minimize the two conflicting objectives of makespan and total cost. The performance of NSPSO algorithm has been evaluated in comparison with NSGA-II algorithm. The simulation results exhibit that NSPSO is a better compromised multi-objective optimization algorithm for workflow grid task scheduling in terms of convergence towards true Pareto optimal front and uniformly distributed solutions with small computation overhead. In future we plan to evaluate the scheduling approach using more than two objectives simultaneously. We will also apply more advanced algorithms like MOPSO [18], 2LB-MOPSO [17], fuzzy dominance based MOPSO [19] etc to obtain the optimal solutions for workflow grid scheduling problem.

References 1. Abraham, A., Liu, H., Zhang, W., Chang, T.G.: Scheduling Jobs on Computational Grids Using Fuzzy Particle Swarm Algorithm, pp. 500–507. Springer, Heidelberg (2006) 2. Attiya, G., Hamam, Y.: Task allocation for maximizing reliability of distributed systems: A simulated annealing approach. Journal of Parallel and Distributed Computing 66, 1259– 1266 (2006) 3. Braun, T.D., Siegal, H.J., Beck, N.: A comparision of Eleven Static Heuristics for Mapping a Class of Independent Tasks onto Heterogeneous Distributed Computing Systems. Journal of Parallel and Distributed Computing 61, 810–837 (2001) 4. Buyya, R., Venugopal, S.: A Gentle Introduction to Grid Computing and Technologies. CSI Communications 29, 9–19 (2005)

190

R. Garg and A.K. Singh

5. Buyya, R., Murshed, M.: GridSim: A Toolkit for Modeling and Simulation of Grid Resource Management and Scheduling, vol. 14, pp. 1175–1220 (2002), http://www.buyya.com/gridsim 6. Chen, W.H., Lin, C.S.: A hybrid heuristic to solve a task allocation problem. Computers & Operations Research 27, 287–303 (2000) 7. Deb, K., Pratap, A., Aggarwal, S., Meyarivan, T.: A Fast Elitist Multi-Objective Genetic Algorithm. In: Deb, K., Rudolph, G., Lutton, E., Merelo, J.J., Schoenauer, M., Schwefel, H.-P., Yao, X. (eds.) PPSN 2000. LNCS, vol. 1917, pp. 849–858. Springer, Heidelberg (2000) 8. Deb, K., Jain, S.: Running Performance Metrics for Evolutionary Multi-objective Optimization. In: Proceedings of Simulated Evolution and Learning (SEAL 2002), pp. 13– 20 (2002) 9. Wieczorek, M., Prodan, R., Fahringer, T.: Scheduling of Scientific Workflows in the ASKALON Grid Environment. SIGMOD 34(3), 56–62 (2005) 10. Haluk, T., Hariri, S., Wu, M.Y.: Performance-Effective and Low-Complexity Task Scheduling for Heterogeneous Computing. IEEE Transactions on Parallel and Distributed Systems 13, 260–274 (2002) 11. Izakian, H., Tork Ladani, B., Zamanifar, K., Abraham, A.: A Novel Particle Swarm Optimization Approach for Grid Job Scheduling. In: Prasad, S.K., Routray, S., Khurana, R., Sahni, S. (eds.) ICISTM 2009. CCIS, vol. 31, pp. 100–109. Springer, Heidelberg (2009) 12. Kennedy, J., Eberhart, R.: Particle Swarm Optimization. In: Proceedings of IEEE International Conference on Neural Networks, vol. 4, pp. 1942–1948 (1995) 13. Li, X.: A Non-dominated Sorting Particle Swarm Optimizer for Multi- objective Optimization. In: Proceeding of Genetic and Evolutionary Computation Conference 2003 (GECCO 20003), Chicago, USA (2003) 14. Ritchie, G., Levine, J.: A fast, effective local search for scheduling independent jobs in heterogeneous computing environments, Technical report, Centre for Intelligent Systems and their Applications, School of Informatics, University of Edinburgh (2003) 15. Subrata, R., Zomaya, Y.A., Landfeldt, B.: Artificial life techniques for load balancing in computational grids. Journal of Computer and System Sciences 73, 1176–1190 (2007) 16. Tsiakkouri, E., Sakellariou, R., Zhao, H., Dikaiakos, M.D.: Scheduling Workflows with Budget Constraints. In: CoreGRID Integration Workshop Pisa, Italy (2005) 17. Zhao, S.Z., Zhao, P.N.: Two-lbests Based Multi-objective Particle Swarm Optimizer. Engineering Optimization 43, 1–17 (2011) 18. Coello, C.A.C., Pulido, G., Lechuga, M.: Handling multi-objective with particle swarm optimization. IEEE Trans. Evol. Comput. 8(3), 256–279 (2004) 19. Praveen, K., Das, S., Welch, S.M.: Multi-Objective Hybrid PSO Using ε-Fuzzy Dominance. In: Proceeding of Genetic and Evolutionary Computation Conference (GECCO 2007), London, UK (2007) 20. Yu, J., Buyya, R.: Scheduling Scientific Workflow Applications with Deadline and Budget constraints using Genetic Algorithms. Scientific Programming Journal 14(1), 217–230 (2006)

Solution of Economic Load Dispatch Problem Using Lbest-Particle Swarm Optimization with Dynamically Varying Sub-swarms Hamim Zafar1, Arkabandhu Chowdhury1, and Bijaya Ketan Panigrahi2 1

Dept. of Electronics and Telecommunication Engg., Jadavpur University, Kolkata, India 2 Senior Member, IEEE, Dept. of Electrical Engineering, IIT Delhi, New Delhi, India

Abstract. This article presents an efficient optimization approach to solve constrained Economic Load Dispatch (ELD) problem using a ‘Lbest-Particle Swarm Optimization with Dynamically Varying Sub-swarms’ (LPSO-DVS). The proposed method is found to give optimal results while working with constraints in the ELD, arising due to practical limitations like dynamic operation constraints (ramp rate limits) and prohibited zones and also accounts valve point loadings. Simulations performed over various systems with different number of generating units with the proposed method have been compared with other existing relevant approaches. Experimental results support the claim of proficiency of the method over other existing techniques in terms of robustness, fast convergence and, most importantly its optimal search behavior.

1

Introduction

Economic load dispatch (ELD) is one of the most important problems involving optimization process in case of electric power system operation. This constrained optimization problem in power systems has the objective of minimizing fuel cost of generating units so as to accomplish optimal generation dispatch among operating units and in return satisfying the system load demand, generator operation constraints with ramp rate limits and prohibited operating zones. Over the years several endeavors have been made to solve this problem, the conventional methods include Lambda iteration method [1][2], base point and participation factors method [1][2]. This non-convex problem is very complex and cannot be solved by the traditional methods, with good result. Recently, different heuristic approaches have been proved to be effective with promising performance. These include evolutionary programming (EP) [3], simulated annealing (SA) [4], genetic algorithm (GA) [5], differential evolution (DE) [6], particle swarm optimization (PSO) [7], etc. The PSO algorithm has been empirically shown to perform well on many optimization problems. However, it may easily get trapped in a local optimum when solving complex multimodal problems. In order to improve PSO’s performance on ELD problems, we present the Lbest-Particle Swarm Optimization with Dynamically Varying Subswarms (LPSO-DVS) utilizing a new learning strategy. B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 191–198, 2011. © Springer-Verlag Berlin Heidelberg 2011

192

2

H. Zafar, A. Chowdhury, and B.K. Panigrahi

Problem Description

Economic Load Dispatch (ELD) problem is one of the different non-linear programming sub-problems of unit commitment. Two alternative models for ELD are detailed below. 2.1

ELD Formulation with Smooth Cost Function

The objective function corresponding to the production cost can be approximated to be a quadratic function of the active power outputs from the generating units. Symbolically, it is represented as NG

Ft cost =  fi (Pi )

Minimize

(1)

i =1

where f i (Pi ) = a i Pi2 + b i Pi + c i ,

i = 1,2,3, ..., N G

(2)

is the expression for cost function corresponding to ithgenerating unit and ai, bi and ci are its cost coefficients. Pi is the real power output (MW) of ithgenerator corresponding to time period t. NG is the number of online generating units to be dispatched. The cost function for unit with valve point loading effect is calculated by using

( (

f i ( Pi ) = a i Pi 2 + bi Pi + ci + ei sin f i Pi min − Pi

))

(3)

Where ei and fi are the cost coefficients corresponding to valve point loading effect. This constrained ELD problem is subjected to a variety of constraints depending upon assumptions and practical implications. These include power balance constraints to take into account the energy balance; ramp rate limits to incorporate dynamic nature of ELD problem and prohibited operating zones. These constraints are discussed as under. Power Balance Constraints or Demand Constraints. This constraint is based on the principle of equilibrium between total system generation

(

NG

 Pi ) and total system i =1

loads (PD) and losses (PL). That is, NG

P

i

= PD + PL

(4)

i =1

where the transmission loss PL is expressed using B- coefficients, given by

PL =

NG NG

NG

i =1 j =1

i =1

  Pi Bij Pj +  B0 i Pi + B00

(5)

Solution of Economic Load Dispatch Problem

193

The Generator Constraints. The power generated by each generator shall be within their lower limit Piminand upper limit Pimax. So that

Pi min ≤ Pi ≤ Pi max

(6)

The Ramp Rate Limits. Under practical circumstances ramp rate limit restricts the operating range of all the online units for adjusting the generator operation between two operating periods. The generation may increase or decrease with corresponding upper and downward ramp rate limits. So, units are constrained due to these ramp rate limits as mentioned below. If power generation increases, Pi − Pi t −1 ≤ URi

(7)

If power generation decreases, Pit −1 −Pi ≤ DRi

(8)

where Pi t −1 is the power generation of unit i at previous hour and URi and DRi are the upper and lower ramp rate limits respectively. The inclusion of ramp rate limits modifies the generator operation constraints (6) as follows.

max(Pi min,URi − Pi ) ≤ Pi ≤ min(Pi max, Pi t −1 − DRi )

(9)

Prohibited Operating Zone. The prohibited operating zones are the range of output power of a generator where the operation causes undue vibration of the turbine shaft. Normally operation is avoided in such regions. Hence, mathematically the feasible operating zones of unit can be described as follows:

  Pi ≤ P pz and Pi ≥ P pz

 pz

 pz

where P and P generating unit i.

(10)

are the lower and upper are limits of a given prohibited zone for

Conflicts in Constraints Handling. If ever, the maximum or minimum limits of generation of a unit as given by (9) lie in the prohibited zone for that generator, then some modifications are to be made in the upper and lower limits for the generator constraints in order to avoid the conflicts. In case, maximum limit for a generator lies in the prohibited zone, the lower limit of the prohibited zone is taken as the maximum limit of power generation for that particular generator. Similarly, care is taken in case the minimum limit of power generation of a generator lies in the prohibited zone by taking upper limit of the prohibited zone as the lower limit of power generation for that generator. 2.2

ELD Formulation with Non-smooth Cost Function

If multiple fuels are used the objective function of an ELD problem is a non-smooth cost function. With multiple fuels, the objective function is a superposition of piecewise quadratic functions

194

H. Zafar, A. Chowdhury, and B.K. Panigrahi

ai1 + bi1 Pi + ci1 Pi 2  2 ai 2 + bi 2 Pi + ci 2 Pi  Fi ( Pi ) = . .  ain + bin Pi + cin Pi 2 

if Pi min ≤ Pi ≤ Pi1 if Pi1 ≤ Pi ≤ Pi 2

(11)

if Pin−1 ≤ Pi ≤ Pi max

where a if , bif , c if are the cost coefficients of generator ‘i’ for the ‘ f th ’ fuel so that the total cost in this case is given by,

Ft cos t =  f i ( Pi )

3

whe re 1 ≤ i ≤ N G

(12)

An Overview of PSO Algorithm

The classical PSO proposed by Kennedy and Eberhart [8] starts with the random initialization of a population of candidate solutions (particles) over the fitness landscape. It works depending on the social behavior of the particles in the swarm. Therefore, it finds the global best solution by simply adjusting the trajectory of each individual towards its own best position and toward the best particle of the entire swarm at each time-step (generation). In a D-dimensional search space, the position  vector of the i-th particle is given by X i = (xi1 , xi2 , , xiD ) and velocity of the i-th  particle is given by Vi = (vi1 , vi2 , , viD ) . Positions and velocities are adjusted and the  objective function to be optimized f ( X i ) is evaluated with the new coordinates at

each time-step. The velocity and position update equations for the d-th dimension of the i-th particle in the swarm may be represented as:

vid = ω * vid + c1 * rand1id *( pbestid − xid ) +c2 * rand 2id *( gbest d − xid ) (13) ,

x = x +v , d i

d i

d i

(14)

where c1 and c2 are the acceleration constants, c1 controls the effect of the personal best position, c2 determines the effect of the best position found so far by any of the particles, rand1id and rand2id are two uniformly distributed random numbers in the range [0, 1]. ω is the inertia weight that balances between the global and local search abilities and takes care of the influence of the previous velocity vector. pbesti = ( pbesti1 , pbesti2 ,, pbestiD ) is the best previous position yielding the best fitness value pbest for the i th particle and gbest = ( gbest 1 , gbest 2 , , gbest D ) is i the best position discovered by the whole population. In the local version of PSO, each particle’s velocity is modified according to its personal best and the best performance achieved so far within its neighborhood instead of learning from the personal best and the best position achieved so far by the whole population in the global version. The velocity updating equation becomes:

Solution of Economic Load Dispatch Problem

195

vid = ω * vid + c1 * rand1id *( pbestid − xid ) + c2 * rand 2id *(lbestid − xid ), (15) here lbest i = (lbest i1 , lbesti2 ,, lbestiD ) is the best position achieved within its neighborhood.

4

Lbest-PSO with Dynamically Varying Sub-swarms

The Lbest-Particle Swarm Optimization with Dynamically Varying Sub-swarms (LPSO-DVS) is a variant of PSO constructed based on the local version of PSO. A special neighborhood topology is used in this variant. Similar kind of topology has been used in DMSPSO [9]. As PSO with smaller neighborhood produces better result LPSO-DVS uses smaller neighborhood. As a result the convergence velocity of the population decreases, diversity increases and better solutions are achieved. The total population is divided into a number of small sized swarms whose own members search for better solution in the search space. In order to slow down the population’s convergence velocity and increase diversity small neighborhoods are used in LPSODVS. Each small sized swarm uses its own members to search for better area in the search space. Since the small sized swarms are searching using their own best historical information, they are easy to converge to a local optimum because of PSO’s convergence property. In order to avoid it we vary the no of particles in each subgroup dynamically to keep co-ordination in the search process of individual swarms. After certain number of iterations new particles are added to a sub-swarm and some of the previous particles go to different sub-swarm and thus the neighborhood of a particle changes dynamically. The sizes of the sub-swarms vary in each generation. The number of particles in a sub-swarm may vary from three to ten. The process of variation of sub-swarm is done randomly after 25 iterations (determined empirically, larger no of iterations does not improve the result much but only increases the complexity). In this way, the information obtained by each swarm is exchanged among the swarms. Particles from different swarms are grouped in a new configuration so that each small swarms search space is enlarged and better solutions are possible to be found by the new small swarms. In the velocity update equation we have used a constriction factor to avoid the unlimited growth of the particles’ velocity. This was proposed by Clerc and Kennedy [10]. Equation 14 becomes

Vi d = χ * (ω *Vi d + c1 * rand1id * ( pbest id − X id ) + c 2 * rand 2 id * (lbest id − X id ))

χ

(16)

is the constriction factor given by

χ = 2 / 2 − c − c 2 − 4c

(17)

c = i ci

(18)

196

5

H. Zafar, A. Chowdhury, and B.K. Panigrahi

Results and Discussions

The applicability and viability of the aforementioned technique for practical applications has been tested on three different power systems consisting of different number of units (6, 15 and 40 units). The obtained results are compared with the reported result of other algorithms. The software has been written in MATLAB-7.5 language and executed on a 2 GHz Intel core 2 duo personal computer with 1024-MB RAM. The parameters used for LPSO-DVS are c1 = c 2 = 1.49455, ω = χ = 0.729 The results of APSO, HS_PSO, Lbest PSO and LPSO_DVS have been taken from fifty individual runs. The results of the other algorithms are taken from the references. 5.1

Six Unit System

The system contains six thermal generating units. The total load demand on the system is 1263 MW. The results are compared with the GA [11], PSO [12], NPSOLRS [13], CPSO [14], BBO [15], SOHPSO [16], ABF-NM [17] methods. Table 1. Comparison of results for 6 unit system Minimum Cost ($/hr.) 15459 15450 15447 15446 15450 15443.8164 15446.02 15443.5751 15444.5734 15443.0963 15443.4815 15442.6901

ALGORITHM GA PSO CPSO1 CPSO2 NPSO_LRS ABF-NM SOHPSO APSO HS_PSO BBO Lbest PSO LPSO_DVS

5.2

Average Cost ($/hr.) 15469 15454 15449 15449 15450.5 15446.95383 15497.35 15449.99 15454.0819 15443.0964 15444.1953 15443.0311

Fifteen Unit System

The system contains fifteen thermal generating units. The total load demand on the system is 2630 MW. The results are compared with the GA [11], PSO [12], CPSO [14], ABF-NM [17] methods for this test system. Table 2. Comparison of results for 15 unit system ALGORITHM GA PSO CPSO1 CPSO2 ABF-NM APSO HS_PSO Lbest PSO LPSO_DVS

Minimum Cost ($/hr.) 33113 32858 32835 32834 32784.50 32742.7774 32692.85714 32742.779594 32692.643046

Average Cost ($/hr.) 33021 33021 32 976.81 32976.6812 32740.1885 32756.9912 32696.164007

Solution of Economic Load Dispatch Problem

5.3

197

Forty Unit System

The system contains forty thermal generating units. The total load demand on the system is 10500 MW. The results are compared with the CEP, FEP, MFEP [18][19], FIA [20], SPSO [21], QPSO [22], and BBO [15] methods for this test system. Table 3. Comparison of results for 40 unit system ALGORITHM CEP FEP MFEP IFEP FIA SPSO QPSO BBO Lbest PSO LPSO_DVS

6

Minimum Cost ($/hr.) 123488.29 122679.71 122647.57 122624.35 121823.80 121787.39 121448.21 121426.95 125191.40 121414.84

Average Cost ($/hr.) 124119.37 124119.37 123489.74 123382.00 122662.48 122474.40 122225.07 121508.0325 125519.6954 121450.3898

Conclusion

This paper has proposed a new type of PSO algorithm, LPSO_DVS based on dividing the total swarm into sub-swarms and dynamically varying the members of the subswarms while finding the local bests and this algorithm is applied to solve constrained economic load dispatch problems. Several non-linear characteristics like ramp rate limits, prohibited operating zones and multiple fuels are also used for practical generator operations. This proposed algorithm has produced results that are better than those generated by other algorithms in the cases discussed; also the convergence characteristics are good. This algorithm has produced better results for both smooth and non smooth cost functions than the other algorithms reported yet. From this limited comparative study, it can be concluded that LPSO_DVS can effectively be used to solve smooth as well as non smooth constrained ELD problems.

References 1. Wood, A.J., Wollenberg, B.F.: Power generation, operation and control. John Wiley & Sons, New York (1984) 2. Chen, C.L., Wang, S.C.: Branch and bound scheduling for thermal generating units. IEEE Trans. Energy Convers 8(2), 184–189 (1993) 3. Yang, H.T., Yang, P.C., Huang, C.L.: Evolutionary Programming based economic dispatch for units with non-smooth fuel cost functions. IEEE Trans. Power Syst. 11(1), 112–118 (1996) 4. Wong, K.P., Fung, C.C.: Simulated annealing based economic dispatch algorithm. Proc. Inst. Elect. Eng. C., Gen., Transm., Distrib. 140(6), 505–519 (1993) 5. Walter, D.C., Sheble, G.B.: Genetic algorithm solution of economic dispatch with valvepoint loading. IEEE Trans. Power Syst. 8(3), 1125–1132 (1993)

198

H. Zafar, A. Chowdhury, and B.K. Panigrahi

6. Coelho, L.S., Mariani, V.C.: Combining of chaotic differential evolution and quadratic programming for economic dispatch optimization with valve-point effect. IEEE Trans. Power Syst. 21(2), 989–996 (2006) 7. Park, J.B., Lee, K.S., Shin, J.R., Lee, K.Y.: Aparticle swarm optimization for economic dispatch with non-smooth cost functions. IEEE Trans. Power Syst. 20(1), 34–42 (2005) 8. Eberhart, R.C., Kennedy, J.: A new optimizer using particle swarm theory. In: Proc. of the Sixth Int. Symposium on Micromachine and Human Science, Nagoya, Japan, pp. 39–43 (1995) 9. Liang, J.J., Suganthan, P.N.: Dynamic multi-swarm particle swarmoptimizer. In: Proc. Swarm Intell. Symp., pp. 124–129 (June 2005) 10. Clerc, M., Kennedy, J.: The particle swarm–explosion, stability, and convergence in a multidimensional complex space. IEEE Trans. Evol. Comput. 6(1), 58–73 (2002) 11. Walter, D.C., Sheble, G.B.: Genetic algorithm solution of economic dispatch with valvepoint loading. IEEE Trans. Power Syst. 8(3), 1125–1132 (1993) 12. Park, J.B., Lee, K.S., Shin, J.R., Lee, K.Y.: A particle swarm optimization for economic dispatch with non-smooth cost functions. IEEE Trans. Power Syst. 20(1), 34–42 (2005) 13. Immanuel Selvakumar, A., Thanushkodi, K.: A new particle swarm optimization solution to nonconvex economic dispatch problems. IEEE Trans. Power Syst. 22(1), 42–51 (2007) 14. Jiejin, C., Xiaoqian, M., Lixiang, L., Haipeng, P.: Chaotic particle swarm optimization for economic dispatch considering the generator constraints. Energy Convers Manage. 48, 645–653 (2007) 15. Bhattacharya, A., Chattopadhyay, P.K.: Biogeography-Based optimization for different economic load dispatch problems. IEEE Trans. Power Syst. 25(2), 1064–1077 (2010) 16. Chaturvedi, K.T., Pandit, M., Srivastava, L.: Self-organizing hierarchical particle swarm optimization for nonconvex economic dispatch problems. IEEE Trans. Power Syst. 23(3), 1079 (2008) 17. Panigrahi, B.K., Ravikumar Pandi, V.: Bacterial foraging optimisation: Nelder–Mead hybrid algorithm for economic load dispatch. IET Gen. Trans. & Distrib. (September 2007) 18. Sinha, N., Chakrabarti, R., Chattopadhyay, P.K.: Evolutionary programming techniques for economic load dispatch. IEEE Trans. Evol. Comput. 7(1), 83–94 (2003) 19. Yao, X., Liu, Y., Lin, G.: Evolutionary programming made faster. IEEE Trans. Evol. Comput. 3(2), 82–102 (1999) 20. Meng, K.: Research of fuzzy self-adaptive immune algorithm and its application, M. E thesis, east China Univ. Sci. Technol., Shanghai, China (2007) 21. Ning, Z.G., Meng, K., Yan, X.F., Qian, F.: An improved particle swarm algorithm and its application in soft sensor modeling. J. East China Univ. Sci. Technol. 33(3), 400–404 (2007) 22. Meng, K., Wang, H.G., Dong, Z.Y.: Quantum-inspired particle swarm optimization for valve-point economic load dispatch. IEEE Trans. Power Syst. 25(1), 215–222 (2010)

Modified Local Neighborhood Based Niching Particle Swarm Optimization for Multimodal Function Optimization Pradipta Ghosh , Hamim Zafar, and Ankush Mandal Dept. of Electronics and Telecommunication Engg., Jadavpur University, Kolkata 700 032, India {iampradiptaghosh,hmm.zafar,ankmd.10}@gmail.com

Abstract. A particle swarm optimization model for tracking multiple peaks over a multimodal fitness landscape is described here. Multimodal optimization amounts to finding multiple global and local optima (as opposed to a single solution) of a function, so that the user can have a better knowledge about different optimal solutions in the search space. Niching algorithms have the ability to locate and maintain more than one solution to a multi-modal optimization problem. The Particle Swarm Optimization (PSO) has remained an attractive alternative for solving complex and difficult optimization problems since its advent in 1995. However, both experiments and analysis show that the basic PSO algorithms cannot identify different optima, either global or local, and thus are not appropriate for multimodal optimization problems that require the location of multiple optima. In this paper a niching algorithm named as Modified Local Neighborhood Based Niching Particle Swarm Optimization (ML-NichePSO)is proposed. The ability, efficiency and usefulness of the proposed method to identify multiple optima are demonstrated using wellknown numerical benchmarks. Keywords: Evolutionary computation, Swarm Intelligence, Multimodal optimization, Niching algorithms, Particle Swarm Optimization, Crowding.

1

Introduction

Multi-objective optimization is one of the leading research fields of modern day. Various algorithms like PSO [1], DE [2], and GA and many more are available for solving single objective optimization problems. But multi-objective problems are not so easy to solve. In case of multi-objective problems the presence of multiple global and local optimal solutions makes it complex. In case of multi-objective problems as there are many peaks and we have to point out all the peaks, it is much more complicated to optimize. Also it will be much better if one algorithm can find all the local optima along with every global optima. However, many practical optimization problems have multiple optima and it is wise to find as many such optima as possible for a number of reasons. Research on solving multimodal problems with EAs dates back to the landmark work of Goldberg and Richardson [3], in which they nicely B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 199–208, 2011. © Springer-Verlag Berlin Heidelberg 2011

200

P. Ghosh, H. Zafar, and A. Mandal

showed the way of introducing niche-preserving technique in a standard Genetic Algorithm (GA) to obtain multiple optimal solutions. Currently the most popular niching techniques used in conjunction with the evolutionary computation community include crowding [4], fitness sharing [3], restricted tournament selection [5], and speciation [6]. Most of existing niching methods, however, possess some difficulties such as: difficulties to pre-specify some niching parameters; difficulties in maintaining discovered solutions in a run; extra computational overhead, and poor scalability when dimensionality is high. The current research on evolutionary multimodal optimization aims at outsmarting these problems by devising more and more efficient optimizers. In this paper we propose a simple yet very powerful hybrid EA that is based on Local Neighborhood based Particle swarm optimization for multimodal optimization. Since its invention in 1995 by Kennedy and Eberhart, PSO [1] is the base of a huge set of evolutionary algorithm. PSO has emerged as a very competitive optimizer for continuous search spaces. The reason for employing the PSO in multimodal optimization process is its high exploring power over the bound-constrained search space and also its ability to search locally. The newly proposed ML-NichePSO has the capability to converge more accurately to the local and global peaks, starting from a uniform initialization in the search space, in comparison to a number of state-of-the-art algorithms for multimodal optimization. We have empirically demonstrated that the performance of the algorithm is insensitive to this parameter provided its values are selected as described later. Empirical formulae for these bounds have also been provided and validated. Finally we have demonstrated the efficacy of the proposed algorithm on a set of multimodal optimization problems that constitute the benchmark.

2

Niching Related Works

2.1

Crowding and Restricted Tournament Selection

Crowding introduced by De Jong [7] allows only a fraction of the population to reproduce and die in each generation. It encourages competition for limited resources among similar individuals in the population. Belonging to this category are crowding, deterministic crowding restricted tournament selection (RTS), and so on. 2.2

Sharing and Clustering

Fitness sharing [8] is based on the concept that a point in a search space has limited resources that need to be shared by any individuals that occupy similar search space behaviors or genetic representations. Sharing in EAs is implemented by scaling the fitness of an individual based on the number of “similar” individuals present in the population. 2.3

Clearing

Clearing (CLR) is an alternative to the sharing methods. Instead of sharing resources between all individuals of a niche as in the fitness sharing scheme, clearing attributes

Modified Local Neighborhood Based Niching Particle Swarm Optimization

201

them only to the best members of each niche and eliminates the other individuals within the same niche. 2.4

Speciation

The concept of speciation depends on radius parameter rs, which measures Euclidean distance from the center of a species to its boundary. All individuals falling within the radius from the species seed are identified as the same species. In addition to the methods listed above, there are niching methods such as sequential niching including de-rating, parallelization, and clustering and others.

3

Overview of the Proposed ML-NichePSO Algorithm

The classical PSO proposed by Kennedy and Eberhart [1] depends on the social interaction between independent agents, here called particles, during their search for the optimum solution using the concept of fitness. After defining the solution space and the fitness function, the PSO algorithm starts by randomly initializing the position and velocity of each particle in the swarm. Each particle in PSO has an adaptable velocity. Moreover, each particle has a memory remembering the best position of the search space that has ever been visited. The velocity matrix is updated according to t t −1 t t −1 t −1 vmn = wvmn + c1U nt 1 ( p mn − xmn ) + c2U nt 2 ( g nt − xmn )

(1)

where the superscripts t and t-1 refer to the time index of the current and the previous iterations,Un1 and Un2 are two uniformly distributed random numbers in the interval [0,1]. The position matrix is updated each iteration according to

X t = X t −1 + V t

(2)

In the local version of PSO each particle’s velocity is adjusted according to its personal best position and the best performance achieved so far within its neighborhood. Then the velocity updating equation is modified as follows. t t −1 t t −1 t t −1 v mn = wv mn + c1U nt 1 ( p mn − x mn ) + c 2U nt 2 (l mn − x mn )

Here l

t mn

3.1

Description of the Proposed Algorithm

(3)

is the best position achieved within its neighborhood after t iterations.

This algorithm is a modified version of PSO with local neighborhood but applied as niching technique. The neighborhood of each particle is selected dynamically (i.e. it can change in every iteration). A particle is selected as neighbor of another particle and vice versa if they satisfy certain conditions.

202

P. Ghosh, H. Zafar, and A. Mandal

1. The first condition is that the distance between them must be less than or equal to some predefined maximum distance ( Dmax ). The Maximum distance is generally equal to one third of the distance between two nearest optima found by the particles till the present iteration. All the particles within a circle or radius Dmax with centre at the position of the particle, whose neighborhood has to be defined, are selected as neighbors of that particle. But initial value of Dmax is selected using a generalized equation for all dimensions given below. D

D max initial = l k

u k

 (x

u k

− x kl ) 2

k =1

10 * Dim

(4) th

where x and x are the lower and upper bounds on the k dimension of the variable vector of Dim dimensions. This empirical equation gives us much better results than the results without any initial definition of Dmax . The value of Dmax is updated in the rest of the iterations as follows.

Dmax = Min (Distances between two optima found so far)/3;

(5)

2. Another selecting criterion is involved in this algorithm. This method provides the dynamic nature of neighborhood. A random number is selected between 1 and 0. If it is 1for a neighborhood’s particle in a particular iteration, then it is the active neighbor of the selected particle for that iteration. Otherwise, the neighbor particle is simply ignored. The grouping of particles is clearly shown in Figure 2. In every iteration each particle compares its local best (lbest) position with its neighbor’s lbest positions and updates its velocity and position using equations 2 and 3 if the above two conditions are met. This updating of velocity and position is performed with probability of 75%. This probability is considered for much more variation in the new particle’s position and velocity, also it can create new particles similar to the crossover vector in DE terminology. Figure 1 presents a simplified block diagram of our proposed algorithm.

Fig. 1. Block diagram of the ML-NichePSO algorithm

Still there are several problems to be solved. In our algorithm we have provided solutions of these problems. They are listed below. A. Some of the particles may move outside the search space during searching operation. In our algorithm, in order to constrain the particles within the search space, we calculate the fitness value of a particle and update its position only if the

Modified Local Neighborhood Based Niching Particle Swarm Optimization

203

particle is within the search space. If a particle moves out of the range it is reinitialized. This feature is not shown explicitly in pseudo code. B. There is always a possibility of crowding of large number of particles in any of the optima. In case of all dimensions, especially high dimensions (Where Dimension is ≥ 10), there is always a high probability of gathering a large proportion of the population in same peak. Due to this reason the probability of finding most of the peaks is reduced. To eradicate this problem in our algorithm we have provided a check whether more than 3 particles are very close to each other or not. We have incorporated a scattering scheme to improve the searching ability. Whenever the aforesaid phenomenon i.e. crowding of a number of particles at the same position occurs this scheme scatters most of the particles to a distant position leaving only few at that location. The scattering of particles can be compared with the physical phenomenon observed in ants when we put something in the location of their gathering except that in our algorithm some particles are remained at the position of gathering. This feature helps to repel particles which are very close or almost same in position and thus helps to increase the probability of finding more peaks as particles are distributed in the search space to find new peaks. As a result of this scattering process the previously found peak is not lost as at least one particle is retained in that position. The scattering is done via following steps. Step1. Check the no of particles located at same position. Two particles will be considered to be at same position iff they satisfy a particular distance criterion. If the distance between two particles is less than 0.5% of Dmax , they will be considered to be at the same position. Step2. Put aside the best 3 particles in terms of fitness for that particular location. Step3. Rest of the particles’ locations are checked dimension wise i.e. whether they are on the lower bound side or upper bound side of the best particle in that location for that particular dimension. Step4. Now the scattering is incorporated using the following formulas. If the particle is on the lower bound side (i.e. within lower bound and best position) for

i th dimension i i i Position updated = Position best − ( Position best − lower _ bound i ) × ( 0.5) × rand ()

(6)

If the particle is on the upper bound side (i.e. within upper bound and best position) for

i th dimension i i i Position updated = Position best + (upper _ bound i − Position best ) × (0 .5) × rand ()

4

Experiments and Results

4.1

Numerical Benchmarks

(7)

The collection of the test functions are mainly composed of 8 multimodal general functions and 7 composition functions of various characteristics, such as irregular

204

P. Ghosh, H. Zafar, and A. Mandal

landscape, symmetric or equal distribution of optima, unevenly spaced optima, etc. A brief description of the functions is given in Table 2. 4.2

Algorithms Compared

Performance of ML-NichePSO is compared with the following standard multimodal evolutionary algorithms: Crowding DE (CDE) [3] , Speciation-based DE (SDE) [8], Fitness-Euclidean distance Ratio PSO (FER-PSO) [9], Speciation-based PSO (SPSO) [10], r2pso [11], r3pso [11], r2psolhc [11], r2psolhc [11], and CMA-ES with Self-Adaptive Niche Radius (S-CMA). 4.3

Performance Measure

The performance of all multimodal algorithms is measured in terms of the following two criteria: 1) Success Rate: The percentage of runs in which all global peaks are successfully found. Table 3 shows the respective data. 2) Average Number of optima found: This is the most important performance measuring criteria of any niching algorithm. All performances are calculated and averaged over 50 independent runs. All the algorithms are implemented in MATLAB 7.5 and executed using a Pentium core 2 duo machine with 2 GB RAM and 2.23 GHz speed. 4.4

Numerical Results

This section presents the experimental results. All the algorithms are run till the maximum number of FEs are exhausted. The second and third column of tables 3 and 4 indicate the level of accuracy and niche radius (for SDE and SPSO) used in the experiments. Values of niche radius values are chosen as recommended by the corresponding authors. The level of accuracy( ε ) is taken too small so that it becomes challenging for the algorithms to locate the peaks. For f7, f8 and all the 10-D composition functions, since it is very difficult to achieve a non-zero success rate, the comparison is based only on the average number of optima found. 4.5

Population and Max_FEs Initialization Table 1. Initialization of population Function Number

Population Size

f1 to f6 f7 f8 CF1 to CF7

50 100 200 100

Maximum number of FEs 10000 20000 40000 50000

Modified Local Neighborhood Based Niching Particle Swarm Optimization

205

Table 2. Benchmark Functions Name

Dim

Test Function

Range

No. of global peaks

f1:Decreasing maxima f2:Uneven maxima

1

x − 0.1 2 º ª f 5 ( x) = exp «− 2 log( 2) ⋅ ( ) ⋅ sin 6 (5π x) 0.8 »¼ ¬

0 ≤ x ≤1

1

f3: Uneven Decreasing Maxima f4: Himmelblau’s function f5: Six-hump camel back

1

1

2 2 1

3 4

f 2 ( x) = sin 6 (5π ( x − 0.05))

0 ≤ x ≤1

5

x − 0.08 2 º ª f 3 ( x ) = exp «− 2 log( 2) ⋅ ( ) ⋅ sin 6 (5π ( x 4 − 0.05)) 0.854 »¼ ¬

0 ≤ x ≤1

1

& 2 f 4 ( X ) = 200 − ( x12 + x 2 − 11) 2 − ( x12 + x 2 − 7) 2

− 4 ≤ x1, x2 ≤ 4

& ª§ º x4 · f5 ( X ) = −4 «¨¨ 4 = 2.2 x12 + 1 ¸¸ x12 + x1 x2 + (−4 + 4 x22 ) x22 » 3¹ ¬© ¼

− 1.9 ≤ x1 ≤ 1.9 − 1.1 ≤ x 2 ≤ 1.1

3

& 1 n f ( X ) = ¦ sin(10 ⋅ log( xi )), n i =1

4 2

f6: 1D inverted Vincent function f7: 2D inverted Vincent function f8: 3D inverted Vincent function CF1

10

Corresponds to CF1 of [12]

−5 ≤ xi ≤ 5

8

CF2

10

Corresponds to CF2 of [12]

−5 ≤ xi ≤ 5

6

CF3

10

Corresponds to CF3 of [12]

−5 ≤ xi ≤ 5

6

CF4

10

Corresponds to CF4 of [12]

−5 ≤ xi ≤ 5

6

CF5

10

Corresponds to CF5 of [12]

−5 ≤ xi ≤ 5

6

CF6

10

Corresponds to CF6 of [12]

−5 ≤ xi ≤ 5

6

CF7

10

Corresponds to CF9 of [12]

−5 ≤ xi ≤ 5

6

2

6

0.25 ≤ xi ≤ 10

36

Where n is the dimensionality of the problem. 3

216

Table 3. Success Rates for test functions Function

ε

f1 f2 f3 f4 f5 f6

0.000001 0.000001 0.000001 0.0005 0.000001 0.0001

r

MLCDE SDE CMA- FERPSO SPSO r2pso r3pso r2psolhc r3psolhc NichePSO ES 0.01 72 100 100 100 100 100 100 100 100 100 0.01 28 60 88 92 88 72 92 92 100 100 0.01 60 100 100 100 100 100 100 100 100 100 0.5 0 72 72 72 0 28 24 28 24 100 0.5 0 60 96 0 56 60 56 52 100 100 0.2 56 48 60 60 72 68 56 52 48 100

206

P. Ghosh, H. Zafar, and A. Mandal Table 4. Average number of peaks found for the test functions

Function

ε

r

f1 f2 f3 f4 f5 f6 f7 f8 CF1 CF2 CF3 CF4 CF5 CF6 CF7

0.000001 0.000001 0.000001 0.0005 0.00001 0.0001 0.001 0.001 0.5 0.5 0.5 0.5 0.5 0.5 0.5

0.01 0.01 0.01 0.5 0.5 0.2 0.2 0.2 1 1 1 1 1 1 1

MLNichePSO 1 5 1 4 2 6 30.6 101 2 2 2 0 2 2 1

a)FEs=100

CDE SDE CMA- FERPSO SPSO r2pso r3pso r2psolhc r3psolhc ES 0.72 1 1 1 1 1 1 1 1 3.96 4.6 4.88 4.92 4.88 4.72 4.92 4.88 5 0.6 1 1 1 1 1 1 1 1 0.32 3.72 3.72 3.68 0.84 2.92 2.76 3 3.12 0.04 2 1.6 1.96 0.08 1.44 1.56 1.56 1.48 5.56 4.88 5.56 5.36 5.6 5.52 5.16 5.36 5.28 23.6 25.72 21.8 22.2 22.52 23.12 33.8 22.8 24.6 68.6 70.12 40.6 45.4 42.2 43.32 152 50.6 75.6 0 1.8 1.08 1.08 0 0 0 0 0 1.2 1.2 1.52 0 0 0 0 0 2 0.72 1.52 0.52 0 0 0 0 0 2.52 0 0 0 0 0 0 0 0 0 1.12 1.32 1.2 0 0 0 0 0 2 0 1.4 1.12 1.2 0 0 0 0 0 0 1.52 0 0 0 0 0 1.8 1.2

b) FEs=500

c) FEs=1000

Fig. 2. Distribution of individual particles in the search space during the evolution of the process for f1

a)FEs=100

b)FEs=500

c) FEs=1000

Fig. 3. Distribution of individuals in the search space during the evolution of the process for f6

4.6

Analysis of Results

The results shown above clearly indicate it has the capability to locate the local peaks satisfactorily other than the global peaks, which should be the target of a multimodal optimization algorithm. In addition, it has also performed efficiently when the

Modified Local Neighborhood Based Niching Particle Swarm Optimization

a)FEs=500

b)FEs=1000

207

c) FEs=5000

Fig. 4. Distribution of individuals in the search space during the evolution of the process for f4

dimensionality is high (D =10). For these high-dimensional composite functions, the traditional PSO variants have completely failed to locate any of the global optima. The algorithm offers a fast convergence towards the results as proved by the figures where almost all the peaks are found in 500FEs only. For two dimensional problems, all the peaks are found after 1000 FEs and after 5000 FEs all the particles are located in any of the global peaks. So this algorithm is time saving, less complex, and efficient in finding the global peaks.

5

Conclusions

In this paper we proposed a multimodal evolutionary optimization technique that integrates concepts of Local Search based Particle Swarm Optimization. Its performance over fifteen Multimodal Benchmark functions are compared with nine state-of-the-art evolutionary multi-modal optimizers based on the performance metrics like success rate, average number of peaks found and peak accuracy. From the Analysis of the results we can firmly propose our algorithm as one of the most powerful niching algorithm present in various publications. Our further work will be to improve our proposed algorithm to use it effectively and efficiently in tracking multiple peaks in a dynamic environment.

References 1. Kennedy, J., Eberhart, R.C.: Particle swarm optimization. In: Proc. IEEE Conf. Neural Networks IV, Piscataway, NJ (1995) 2. Storn, R., Price, K.: Differential evolution a simple and efficient heuristic for global optimization over continuous spaces. J. Global Optimization 11(4), 341–359 (1997) 3. Goldberg, D.E., Richardson, J.: Genetic algorithms with sharing for multimodal function optimization. In: Proceedings of the Second International Conference on Genetic Algorithms, pp. 41–49 (1987) 4. Thomsen, R.: Multimodal optimization using Crowding-based differential evolution. In: Proceedings of the IEEE 2004 Congress on Evolutionary Computation, pp. 1382–1389 (2004) 5. Harik, G.R.: Finding multimodal solutions using restricted tournament selection. In: Proceedings of the 6th International Conference on Genetic Algorithms, San Francisco, pp. 24–31 (1995)

208

P. Ghosh, H. Zafar, and A. Mandal

6. Petrowski, A.: A clearing procedure as a niching method for genetic algorithms. In: Proc. of 3rd IEEE Congress on Evolutionary Computation, pp. 798–803 (1996) 7. De Jong, K.A.: An analysis of the behavior of a class of genetic adaptive systems, Doctoral Dissertation, University of Michigan (1975) 8. Kennedy, J.: Small worlds and mega-minds: effects of neighborhood topology on particle swarm performance. In: Proc. of IEEE Congress on Evolutionary Computation (CEC 1999), Piscataway, NJ, pp. 1931–1938 (1999) 9. Li, X.: Multimodal function optimization based on fitness-euclidean distance ratio. In: Proc. Genet. Evol. Comput. Conf., pp. 78–85 (2007) 10. Li, X.: Adaptively Choosing Neighbourhood Bests Using Species in a Particle Swarm Optimizer for Multimodal Function Optimization. In: Deb, K., et al. (eds.) GECCO 2004. LNCS, vol. 3102, pp. 105–116. Springer, Heidelberg (2004) 11. Li, X.: Niching without niching parameters: particle swarm optimization using a ring topology. IEEE Transaction on Evolutionary Computation 14(1) (February 2010) 12. Qu, B.Y., Suganthan, P.N.: Novel multimodal problems and differential evolution with ensemble of restricted tournament selection. In: IEEE Congress on Evolutionary Computation, Barcelona, Spain (July 2010) 13. Qu, B.Y., Suganthan, P.N., Liang, J.J.: Differential Evolution with Neighborhood Mutation for Multimodal Optimization. IEEE Trans. on Evolutionary Computation, doi:10.1109/TEVC.2011.2161873 14. Das, S., Maity, S., Qu, B.-Y., Suganthan, P.N.: Real-parameter evolutionary multimodal optimization — A survey of the state-of-the-art. Swarm and Evolutionary Computation 1(2), 71–88 (2011)

Constrained Function Optimization Using PSO with Polynomial Mutation Tapas Si1, Nanda Dulal Jana1, and Jaya Sil2 1

Department of Information Technology, National Institute of Technology, Durgapur, West Bengal, India 2 Department of Computer Science and Technology, BESU, West Bengal, India {c2.tapas,nanda.jana}@gmail.com [email protected]

Abstract. Constrained function optimization using particle swarm optimization (PSO) with polynomial mutation is proposed in this work. In this method nonstationary penalty function approach is adopted and polynomial mutation is performed on global best solution in PSO. The proposed method is applied on 6 benchmark problems and obtained results are compared with the results obtained from basic PSO. The experimental results show the efficiency and effectiveness of the method.

1 Introduction Constrained Optimization (CO) problems are encountered in numerous applications. Structural optimization, engineering design, VLSI design, economics, allocation and location problems are just few of the scientific fields in which CO problems are frequently met [1]. The CO problem can be represented as the following nonlinear programming problem: ( ),

,

=

,…,

(1)

Subject to: ( )

0, = 1,2, … ,

( ) = 0, =

+ 1, … ,

(2) (3)

Usually equality constraints are transformed into inequalities of the form ( )

0,

=

+ 1, … ,

(4)

A solution X is regarded as feasible if gi(X) +b

(2)

being x the example to be classified. In the linearly separable case, learning the maximal margin hyperplane (w, b) can be stated as a convex quadratic optimization problem with a unique solution: minimize ||w||, subject to the constraints (one for each training example): yi (< w.xi > +b) ≥ 1 (3) The SVM model has an equivalent dual formulation, characterized by a weight vector α and a bias b. In this case, α contains one weight for each training vector, indicating the importance of this vector in the solution. Vectors with non null weights are called support vectors. The dual classification rule is: f (x, α, b) =

N 

yi αi < xi .x > +b

(4)

i=1

The α vector can be calculated also as a quadratic optimization problem. Given the optimal α∗ vector of the dual quadratic optimization problem, the weight vector w∗ that realizes the maximal margin hyperplane is calculated as: ∗

w =

N 

yi α∗i xi

(5)

i=1

The b∗ has also a simple expression in terms of w∗ and the training examples (xi , yi )i=1 . The advantage of the dual formulation is that efficient learning of non-linear SVM separators, by introducing kernel functions. Technically, a kernel function calculates a dot product between two vectors that have been (non linearly) mapped into a high dimensional feature space. Since there is no need to perform this mapping explicitly, the training is still feasible although the dimension of the real feature space can be very high or even infinite. N

428

S. Saha, A. Ekbal, and S. Saha

By simply substituting every dot product of xi and xj in dual form with any kernel function K(xi , xj ), SVMs can handle non-linear hypotheses. Among the many kinds of kernel functions available, we will focus on the d-th polynomial kernel: K(xi , xj ) = (xi .xj + 1)d Use of d-th polynomial kernel function allows us to build an optimal separating hyperplane which takes into account all combination of features up to d. We develop our system using SVM [4,7] which perform classification by constructing an N-dimensional hyperplane that optimally separates data into two categories. We have used YamCha4 toolkit, an SVM based tool for detecting classes in documents and formulating the gene mention detection task as a sequential labeling problem. We use TinySVM-0.075 classifier for classification and the polynomial kernel function. 2.2 Named Entity Features Feature selection plays an important role for the success of machine learning techniques. We came up with a variety of features for constructing various models based on the SVM framework. Most of these features are easy to derive and don’t require deep domain knowledge and/or external resources for their generation. Due to the use of variety of features, the individual classifiers achieve very high accuracies. i+3 1. Context words: These are the words occurring within the context window wi−3 = i+2 i+1 wi−3 . . . wi+3 , wi−2 = wi−2 . . . wi+2 and wi−1 = wi−1 . . . wi+1 , where wi is the current word. This feature is considered with the observation that surrounding words carry effective information for identification of NEs. 2. Word prefix and suffix: These are the word prefix and suffix character sequences of length up to n. The sequences are stripped from the leftmost (prefix) and rightmost (suffix) positions of the words. We set the feature values to ‘undefined’ if either the length of wi is less than or equal to n − 1, wi is a punctuation symbol or if it contains any special symbol or digit. We experiment with n=3 (i.e., 6 features) and 4 (i.e., 8 features) both. 3. Word length: We define a binary valued feature that fires if the length of wi is greater than a pre-defined threshold. Here, the threshold value is set to 5. This feature captures the fact that short words are likely not to be NEs. 4. Infrequent word: A list is compiled from the training data by considering the words that appear less frequently than a predetermined threshold. The threshold value depends on the size of the dataset. Here, we consider the words having less than 10 occurrences in the training data. Now, a feature is defined that fires if wi occurs in the compiled list. This is based on the observation that more frequently occurring words are rarely the NEs. 5. Part of Speech (PoS) information: POS information is a critical feature for NERC. In this work, we use POS information of the current and/or the surrounding token(s) as the features. This information is available with the data. 4 5

http://chasen-org/ taku/software/yamcha/ http://cl.aist-nara.ac.jp/ taku-ku/software/TinySVM

A Supervised Approach for Gene Mention Detection

429

6. Dynamic feature: Dynamic feature denotes the output tags ti−3 ti−2 ti−1 , ti−2 ti−1 , ti−1 of the word wi−3 wi−2 wi−1 , wi−2 wi−1 , wi−1 preceding wi in the sequence w1n . 7. Word normalization: We define two different types of features for word normalization. The first type of feature attempts to reduce a word to its stem or root form. This helps to handle the words containing plural forms, verb inflections, hyphen, and alphanumeric letters. The second type of feature indicates how a target word is orthographically constructed. Word shapes refer to the mapping of each word to their equivalence classes. Here each capitalized character of the word is replaced by ‘A’, small characters are replaced by ‘a’ and all consecutive digits are replaced by ‘0’. For example, ‘IL’ is normalized to ‘AA’, ‘IL-2’ is normalized to ‘AA-0’ and ‘IL-88’ is also normalized to ‘AA-0’. 8. Orthographic features: We define a number of orthographic features depending upon the contents of the wordforms. Several binary features are defined which use capitalization and digit information. These features are: initial capital, all capital, capital in inner, initial capital then mix, only digit, digit with special character, initial digit then alphabetic, digit in inner. The presence of some special characters like (‘,’,‘-’,‘.’,‘)’,‘(’ etc.) is very much helpful to detect NEs, especially in biomedical domain. For example, many biomedical NEs have ‘-’ (hyphen) in their construction. Some of these special characters are also important to detect boundaries of NEs. We also use the features that check the presence of ATGC sequence and stop words. The complete list of orthographic features is shown in Table 1. Table 1. Orthographic features Feature InitCap InCap DigitOnly DigitAlpha Hyphen CapsAndDigits StopWord AlphaDigit GreekLetter

Example Src mAb 1, 123 2× NFkappaB, 2A 32Dc13 at, in p50, p65 alpha, beta

Feature AllCaps CapMixAlpha DigitSpecial AlphaDigitAlpha CapLowAlpha RomanNumeral ATGCSeq DigitCommaDigit LowMixAlpha

Example EBNA, LMP NFkappaB, EpoR 12-3 IL23R, EIA Src, Ras, Epo I, II CCGCCC, ATAGAT 1,28 mRNA, mAb

3 Datasets, Evaluation Results and Discussions In this section we present the details of datasets and evaluation results. 3.1 Datasets and Evaluation Technique We evaluate our proposed approach with the GENETAG training and test datasets, available at the site 6 . GENETAG covers a more general domain of PubMed. It contains 6

ftp://ftp.ncbi.nlm.nih.gov/pub/tanabe/GENEATG.tar.gz

430

S. Saha, A. Ekbal, and S. Saha

Table 2. Evaluation results on GENETAG datasets (training: GENETAG, test: GENETAG) with various feature subsets. Here, the following abbreviations are used: ’CW’:Context words, ’PS’: Size of the prefix, ’SS’: Size of the suffix, ’WL’: Word length, ’IW’: Infrequent word, ’NO’: Normalization feature, ‘PoS’: PoS information, ’OR’: Orthographic feature, ‘Dyn’: Dynamic feature, [-i, j]: context words spanning from the left ith position to the jth right position, All[i,j]: All feature combinations within the context except dynamic NE for the left ith positions, X: Denotes the presence of the corresponding feature, ‘r’: recall, ‘p’: precision, ‘F’: F-measure (we report percentages). Classi fiers SV M1 SV M2 SV M3 SV M4 SV M5 SV M6 SV M7 SV M8

CW

PS SS WL IW NO OR POS

Dyn r

p

F

[-1,+1] [-2,+2] [-2,+2] [-2,+2] [-2,+2] [-2,+2] [-2,+2] [-3,+3]

4 4 4 4 3 3 3 4

-1 -2 -2 -2 -2 -2 -2 -3

91.51 93.21 93.36 93.50 93.14 93.32 93.29 93.14

92.54 93.74 93.65 93.95 93.64 93.69 93.69 93.65

4 4 4 4 3 3 3 4

X X X X X X X X

X X X X X X X X

X X X X X X X X

X X X X X X X X

X X [-2,+2] [-2,0] X [-2,+2] [-2,0] X

93.60 94.27 94.20 94.41 94.14 94.07 94.10 94.17

both true and false gene or protein names in a variety of contexts. In GENETAG, not all the sentences of abstracts were included, rather more NE informative sentences were considered. GENETAG selected longer text fragments as entity reference. GENETAG also includes the semantic category word ‘protein’ for protein annotation. GENETAG is more inclined to select more descriptive expressions for protein annotations. During annotations of GENETAG corpus, some semantic constraints were chosen to make sure that tagged entities must contain their true meanings in the sentence contexts. Based on the gene names from GenBank7 , the GENETAG corpus includes domains, complexes, subunits, and promoters when the annotated entities refer to specific genes/proteins. Gene mentions in both the training and test datasets were annotated with the ‘NEWGENE’ tag and the overlapping gene mentions were distinguished by another tag ‘NEWGENE1’. However, in this work, we use the standard BIO notations (as in GENIA corpus) to properly denote the boundaries of gene names, and we replace all the ‘NEWGENE1’ tags by ‘NEWGENE’ for training and testing. The training dataset contains 7,500 sentences with 8,881 gene mentions. The average length per protein (or, gene) mention is 2.1 tokens. The test dataset consists of 2,500 sentences with 2,986 gene mentions. The system is evaluated using the evaluation script that was provided by the BioCreative-II 8 evaluation challenge for the gene mention detection task. All the classifiers are evaluated in terms of recall, precision and F-measure. The definitions of recall and precision are given below: Number of correctly found NE chunks recall = Number of NE chunks in the gold standard test data Number of correctly found NE chunks and precision = Number of identified NE chunks . 7 8

http://www.ncbi.nlm.nih.gov/Genbank/ http://www.biocreative.org/news/biocreative-ii/

A Supervised Approach for Gene Mention Detection

431

The value of the metric F-measure, which is the weighted harmonic mean of recall and precision, is calculated as below: Fβ =

(1 + β 2 )(recall + precision) , β=1 β 2 × precision + recall

These are the modified versions of the CoNLL-2003 shared task [6] evaluation script. The script outputs three sets of F-measures according to exact boundary match, right and left boundary matching. In the right boundary matching only right boundaries of entities are considered without matching left boundaries and vice versa. We use the same strict matching criterion that was followed in the Biocreative-II shared task evaluation script 9 . 3.2 Results and Discussions Results of the individual models are reported in Table 2. Each of these models is trained with the subset of features as described in Subsection 2.2. The highest performance corresponds to SV M4 classifier that yields the overall recall, precision and F-measure values of 94.41%, 93.50%, and 93.95%, respectively. The consistent good performance in each of the models shows the efficacy of the effective feature sets that we identified. In order to obtain higher performance, we combine various SVM based models using the following two voting techniques. – Majority Voting : In this model, all the individual classifiers are combined together into a final system based on the majority voting of the output class labels. If all the outputs differ then anyone is selected randomly. – Weighted Voting : This is a weighted voting approach. In each classifier, weights are calculated based on the average F-measure value of the 3-fold cross validation on the training data. The majority voted technique achieves the overall recall, precision and F-measure values of 94.61%, 94.10%, and 94.35%, respectively. The weighted vote based technique provides the overall recall, precision and F-measure values of 94.95%, 94.32% and 94.63%, respectively. Results show that the voted approach achieves the performance level, higher than the individual classifiers. To the best of our knowledge this is so far the highest performance in comparison to any previously developed systems that made use of the same datasets. Table 3. Overall evaluation results on GENETAG datasets (training: GENETAG, test: GENETAG) we report percentages) Classification Scheme Best individual classifier Majority Voted Approach Weighted Voted Approach

9

recall 94.41 94.61 94.95

precision 93.50 94.10 94.32

F-measure 93.95 94.35 94.63

http://www-tsujii.is.s.u-tokyo.ac.jp/GENIA/ERtask/report.html

432

S. Saha, A. Ekbal, and S. Saha

4 Conclusion and Future Works In this paper we have proposed a voted approach for gene mention detection task. We have used SVM framework as the base classifiers to generate different classification models by varying the available features. We came up with a very rich feature set that itself can achieve very high accuracy. The most important characteristics of our system is that all the features are mostly identified and developed without using any deep domain knowledge and/or external resources. Evaluation results with the benchmark datasets of GENTAG corpus showed the overall recall, precision and F-measure values of 94.41%, 93.50%, and 93.95%, respectively. Final recall, precision and F-measure values after applying the weighted vote based ensemble technique on the SVM based models are 94.95%, 94.32% and 94.63%, respectively. This shows that our approach achieves the state-of-the-art performance. In future we would like to apply some other statistical machine learning algorithms like conditional random field (CRF) and maximum entropy (ME) for solving this gene mention task. Future work also includes use of some new ensemble techniques for combing the outputs of classifiers.

References 1. Aronson, A.R., Bodenreider, O., Chang, H.F., Humphrey, S.M., Mork, J.G., Nelson, S.J., Rindflesch, T.C., Wilbur, W.J.: The NLM Indexing Initiative. In: Proceedings of 2000 AMIA Annual Fall Symposium (2000) 2. Finkel, J., Dingare, S., Manning, C., Nissim, M., Alex, B., Grover, C.: Exploring the boundaries: gene and protein identification in biomedical text. BMC Bioinformatics 6 (2005) 3. Hirschman, L., Yeh, A., Blaschke, C., Valencia, A.: Overview of BioCreAtIvE: critical assessment of information extraction for biology. BMC Bioinformatics 6 (2005) 4. Joachims, T.: Making Large Scale SVM Learning Practical, pp. 169–184. MIT Press, Cambridge (1999) 5. Taira, H., Haruno, M.: Feature Selection in SVM Text Categorization. In: Proceedings of AAAI 1999 (1999) 6. Tjong Kim Sang, E.F., De Meulder, F.: Introduction to the Conll-2003 Shared Task: Language Independent Named Entity Recognition. In: Proceedings of the Seventh Conference on Natural Language Learning at HLT-NAACL 2003, pp. 142–147 (2003) 7. Vapnik, V.N.: The Nature of Statistical Learning Theory. Springer-Verlag New York, Inc. (1995)

Incorporating Fuzzy Trust in Collaborative Filtering Based Recommender Systems Vibhor Kant and Kamal K. Bharadwaj School of Computer and Systems Sciences, Jawaharlal Nehru University, New Delhi, India {vibhor.kant,kbharadwaj}@gmail.com

Abstract. Collaborative filtering based recommender system (CF-RS) provides personalized recommendations to users utilizing the experiences and opinions of their nearest neighbours. Although, collaborative filtering (CF) is the most successful and widely implemented filtering, data sparsity is still a major concern. In this work, we have proposed a fuzzy trust propagation scheme to alleviate the sparsity problem. Since trust is often a gradual trend, so trust to a person can be expressed more naturally using linguistic expressions. In this work, fuzzy trust is represented by linguistic terms rather than numerical values. We discuss the basic trust concepts such as fuzzy trust modeling, propagation and aggregation operators. An empirical evaluation of the proposed scheme on well known MovieLens dataset shows that fuzzy trust propagation allows reducing the sparsity problem of RSs while preserving the quality of recommendations. Keywords: Recommender System, Collaborative Filtering, Trust Network, Trust Propagation, Fuzzy Trust.

1

Introduction

The explosive growth of e-commerce and web-based services has made the question of the exploration of information and selection progressively more serious; users are overloaded by options to judge and they may not have the time or knowledge to evaluate these options personally. Recommender systems (RSs) have been proposed to suggest relevant items for online users to cope with the information overload and have become one of the most popular tools in electronic commerce and web-based services [1] [3]. Recommender systems employ mainly three information filtering techniques, collaborative filtering, content-based filtering (CBF) and hybrid filtering techniques [1]. CF is the most widely used technique for web-based RSs. A CF system searches for similar users and then uses ratings from this set of users to predict items that will be liked by the current user [1][9]. Although CF has been used effectively in a wide range of applications, data sparsity is still a major concern. Recent research on trust aware RSs [9] has shown that they are more robust against sparsity and are more capable of making recommendations. There exist several algorithms for computing, propagating and aggregating the trust in the trust network B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 433–440, 2011. © Springer-Verlag Berlin Heidelberg 2011

434

V. Kant and K.K. Bharadwaj

[4][5][6][7][8][10][11][12] etc. Golbeck [5] described a comprehensive review of trust on the Web and proposed algorithms for trust propagation. Josang [7] developed trust propagation operators on the basis of Subjective Logic. Guha et. al., [6] was the first one who utilized the idea of transitivity of distrust and developed a framework for trust propagation [11][12]. Sometimes crisp modeling of trust is not enough for inferring accurate information especially in contradictory situation and also people naturally use linguistic expressions rather than numeric values to describe their trust. Fuzzy sets seem to be an ideal choice for trust modeling. Here, we proposed a fuzzy trust based CF recommender system that suggests items to users in an effective way. The main contribution of our work is fuzzy trust modeling, propagation and aggregation operators. We presented an experimental analysis on MovieLens dataset and show that our work does alleviate the sparsity problem associated with traditional CF. The rest of this paper is organized as follows: Section 2 provides the fuzzy trust modeling, while section 3 describes the overall framework of our proposed work. Our experimental analysis is presented in section 4 and last section concludes the paper with future work.

2

Fuzzy Trust Modeling

Generally, in trust networks, users specify their trusts to other persons in numerical values. For example, in seven scale ratings, for pessimistic users, a rating of 7 may mean highly trusted but for optimistic users it may mean somewhat high trusted. Is the difference between 2 and 3 the same as the difference 5 and 6? These all contribute to fuzziness that arises from the subjective nature of human. Therefore, trust is treated as a fuzzy variable so there is a need to develop gradual models of trust that quantify the degree to which users may trust each other. We quantified trust into seven triangular fuzzy numbers extremely low trust (ELT), very low trust (VLT), low trust (LT), average trust (AT), high trust (HT), very high trust (VHT) and extremely high trust (EHT) as shown in Fig. 1, with the following membership functions: TELT x T Here a i

x

2 0,

0, x i 2, i x,

x,

x x x

i

2 2

i 2 i

x 1

(1)

2, x i x

i 1 i

(2)

VLT, LT, AT, HT, VHT and EHT for i=3, 4, 5, 6 and 7 respectively. TEHT x

0, x 6,

x 6

6 7

(3)

Fuzzy trust network is shown in Fig. 2. in which users describe trust ratings to each other employing linguistic terms.

Incorporating Fuzzy Trust in Collaborative Filtering Based Recommender Systems

435

Membership value

1.5

1

ELT

VLT

LT

AT

HT

VHT

4

5

6

EHT

0.5

0

1

2

3

7

Trust levels

Fig. 1. Membership functions for trust

Fig. 2. Fuzzy trust network

3

Incorporation of Fuzzy Trust in CF-RS

In this section, we proposeed fuzzy trust propagation and aggregation operators and developed fuzzy trust based d recommendations framework. 3.1

Fuzzy Trust Propag gation and Aggregation Operators

Propagation operators are used u to handle the problem of establishing trust informattion in unknown user using direectly connected trusted users for current user. If a trustts b and b trusts c, this informattion is used to establish trust in c for a. The only problem m in trust propagation is that thee propagation trust is not perfectly transitive. For exam mple if a trusts b completely and d b trusts c completely, then it has no evidence that a truusts c completely. Hence trust propagation p can be considered as a certain degree of flucctuation [10]. Propagation and aggregation operators for fuzzy trust are as follows: Max-Min Operator S oS

x, z , max min µS x, y , µS y, z

|x X, y Y, z Z

(4)

Max-Mean Operator S oS

x, z , max 1 2

µS x, y

µS y, z

x X, y Y, z Z

(5)

436

V. Kant and K.K. Bharadwaj

where S and S are the fuzzy relations on X Y and Y Z respectively. If we implement the computation of trust with max-min operator then propagation utilizes the MIN operator and aggregation the MAX operator. Proposed Operators. In this subsection, a fuzzy trust model is proposed to evaluate trust ratings between two nodes as source and sink. This model consists of two algorithms, one algorithm for propagation of fuzzy trust and another for aggregation. The proposed technique is hereafter referred to as New-Op. Propagation Operator. This operator is fuzzification of the operator proposed by [10] which is based upon the certain percentage of fluctuation of the rating. In our trust network, the rating of trust from one node to other node is assigned to a linguistic variable in the range of [ELT, EHT]. If a person (node) does not know another, then the operator tries to find out estimated trust rating using a chain of their friends. In each chain, the last person knows the trustee directly. Then the trust rating for trustee depends upon the last rating. But this rating is also influenced by the other intermediary nodes. If these nodes are trustworthy, the source may accept the last rating with high confidence. The main steps involved in this operator for each path in the trust network are as follows: (a) The suggested rating (R) is the last rating in the trust network. ) among other intermediary nodes except last (b) Compute minimum rating ( rating using Zadeh’s extension principle [13] , and of fuzzy sets EHT, ELT and (c) Compute heights respectively[13] (d) Since rating R may be too exaggerated for optimistic users or too conservative for pessimistic users, therefore compute the amount of fluctuation in R is as follows: For optimistic users, probability of fluctuation in R is 0 and for pessithen on average, mistic user the fluctuation will be at most absolutely Probability of fluctuation and error in R are computed as pr Fluct

(6) EHT R

error

(7)

(e) If R AT then R R error otherwise (f) The estimated trust is given by Trust

pr Fluct

R

R 1

R

error

pr Fluct

(8) R

(9)

(g) Compute similarity between Trust (9) and each of the linguistic sets defined in section 2. Since all these sets are triangular fuzzy numbers so similarity between two triangular fuzzy numbers a and b is given by sim a, b where

,

,

&

∑

1 ,

,

|

|

(10)

Incorporating Fuzzy Trust in Collaborative Filtering Based Recommender Systems

437

(h) Final trust rating between two nodes is the most similar one to estimated Trust. Aggregation Operator. If there are several paths between source and sink such as shown in Fig.2, then it is a need to have a method for aggregating. We used the most common operator MAX [13] for aggregating the trusted ratings from different paths. An example. We show how to compute the final trust rating between node 1 and node 6 shown in Fig 2 using New-Op operator. There are two paths from node 1 to node 6: 1-2-3-6 and 1-4-5-6 For the First path, the suggested rating is R=ELT (1, 1, 2 ) using step(a) ; = EHT (6, 7, 7) using MIN operator [13] in step(b); =7, =1, and =7 be the heights of EHT, ELT and ; Pr =0 and error = (0,0,0) using = (1,1,2) and Trust= (1,1,2) from steps (e) and (f). after it, the trust step (d); Now rating, the most similar one linguistic set to the (1,1,2) is ELT using formula(10). Similarly, for the second path, the trust rating between node 1 and node 6 is LT. Aggregation can be done using MAX operator [13]. Output of this algorithm, the final trust rating between node 1 and node 6 is LT. 3.2

Fuzzy Trust Based Recommendation Framework

The overall system takes as input the trust matrix or the rating matrix and outputs a matrix of predicted ratings, then the RS selects from the row of predictive ratings relative to the user. Firstly, we apply the traditional CF algorithm on MovieLens dataset. In order to make recommendations to the active users, CF performs three steps: • • •

It compares the active user’s ratings with other users in the dataset. It can be done by computing similarity between users using Pearson formula[1] It predicts ratings for unseen items to the active user on the basis of the most similar users (neighbours). [1] It recommends items to the user with highest predictive ratings

Secondly, we incorporate fuzzy trust in the above CF-RS to deal with sparsity problem. Due to data sparsity, in many cases neighbourhood size is very small and therefore accurate predictions cannot be done. When sparsity level is high, sometimes the neighbourhood set can be just empty and no predictions would be possible. We apply fuzzy trust propagation as discussed above in subsection in 3.1, to enlarge the neighbourhood size that results in better recommendations. At last, the predicted rating of item i for the active user is the weighted sum of the ratings given to item i by n neighbours using Resnick formula [1] ∑ ,

,

∑

, ,

(11)

Here neighbours can be taken from the user similarity matrix generated by Pearson formula or from the estimated trust matrix generated by propagation operators.

438

4

V. Kant and K.K. Bharadwaj

Experimental Results

To demonstrate the effectiveness of proposed operators of fuzzy trust, we conducted experiments on the popular MovieLens dataset. 4.1

Design of Experiments

The MovieLens dataset consists of 100,000 ratings (1-5) provided by 943 users on 1682 movies; and each user has rated at least 20 and at most 737 movies. For our experiments we choose three subsets from the data, containing 50,100 and 150 users called ML50, ML100 and ML150 respectively. Since there is no availability of trust statements in MovieLens dataset, we randomly generate the trust matrix consisting linguistic ratings for these datasets. This is to illustrate the effectiveness of the proposed work under varying number of participating users. Each of the datasets was randomly split into 60% training data and 40% test data. The ratings of the items in test data are considered as items unseen by the active user, while the ratings in the training set is used for neighbourhood construction. In order to test the performance of our scheme, we measure system accuracy using the mean absolute error (MAE) and coverage of the system. MAE measures the average absolute deviation of predicting rating of an item from the actual rating for the item. Coverage is measured as number of items for which RSs can generate prediction over total number of unseen items. 4.2

Results

To demonstrate the ability of the proposed scheme New-Op-CF to offer better accuracy we compare the MAE and coverage with CF, Max-Min-CF, and Max-Mean-CF. The results are presented in Table 1. The MAE is computed based on the average over 20 runs of the experiments over different datasets. A lower value of MAE corresponds to a better performance. It is clear from table 1 the proposed scheme New-OpCF considerably outperforms other methods. Table 1. MAE comparison of New-Op-CF with CF, Max-Min-CF and Max- Mean-CF

Datasets ML50 MAE ML100 MAE ML150 MAE

CF 0.8960 0.9194 0.8853

Max-Min-CF 0.8735 0.9066 0.8888

Max-Mean-CF 0.8735 0.9065 0.8887

New-Op-CF 0.8744 0.8930 0.8668

The total coverage for New-Op-CF is always greater than the other techniques shown in Table 2. A higher value of coverage implies the better performance of the proposed scheme. The MAE for the different runs of the experiment for ML50 is shown in Fig. 3. A total of 20 runs were made for all datasets. The proposed method New-Op-CF performed better than any of the other techniques in the terms of predictive accuracy as well as coverage.

Incorporating Fuzzy Trust in Collaborative Filtering Based Recommender Systems

439

Table 2. Comparison of Coverage of New-Op-CF with CF, Max-Min-CF and Max-Mean-CF

Datasets ML50 Coverage ML100 Coverage ML150 Coverage

CF 86.7 87.7 88.8

Max-Min-CF 87.2 87.9 89.2

Max-Mean-CF 87.4 87.9 90.2

New-Op-CF 88.6 88.5 90.5

0.91

Proposed

Maxmin

Maxmean

CF

0.9

0.89

0.88

0.87

0.86

0.85

0

2

4

6

8

10

12

14

16

18

20

Fig. 3. MAE for ML50 over 20 runs

5

Conclusions and Future Work

This research proposes modeling of fuzzy trust and corresponding propagation operators to deal with the sparsity problem. Incorporation of fuzzy trust network among users has resulted in alleviating data sparsity problem, thereby producing quality recommendations. Through experimental results, we have clearly demonstrated that New-Op-CF outperforms traditional CF, Max-Min-CF and Max Mean-CF. Our future works goes in several directions. First, we will focus on developing different efficient techniques to compute and propagate fuzzy trust [11][12]. Second, it is to be seen how two level filtering based on both trust and reputation [4] and utilization of various sparsity measures [2] can be incorporated in the fuzzy trust enhanced CF system to further improve its performance.

References 1. Adomavicius, G., Tuzhilin, A.: Toward The Next Generation of Recommender Systems: A Survey of The State-of-The-Art and Possible Extensions. IEEE Trans. Knowledge and Data Engineering 17(6), 734–749 (2005) 2. Anand, D., Bharadwaj, K.K.: Utilizing Various Sparsity Measures for Enhancing Accuracy of Collaborative Recommender Systems Based on Local and Global Similarities. Expert Systems with Applications 38(5), 5101–5109 (2010)

440

V. Kant and K.K. Bharadwaj

3. Al-Shamri, M.Y.H., Bharadwaj, K.K.: Fuzzy-Genetic Approach to Recommender System Based on A Novel Hybrid User Model. Expert Systems with Applications 35(3), 1386– 1399 (2008) 4. Bharadwaj, K.K., Al-Shamri, M.Y.H.: Fuzzy Computational Models for Trust and Reputation Systems. Electronic Commerce Research and Applications 8(1), 37–47 (2009) 5. Golbeck, J.: Trust and Nuanced Profile Similarity in Online Social Networks. ACM Transactions on the Web (TWEB) 3(4), 1–33 (2009) 6. Guha, R., Kumar, R., Raghavan, P., Tomkins, A.: Propagation of Trust and Distrust. In: Proceedings of the 13th International Conference on World Wide Web, New York, pp. 403–412 (2004) 7. Jøsang, A., Marsh, S., Pope, S.: Exploring Different Types of Trust Propagation. In: Stølen, K., Winsborough, W.H., Martinelli, F., Massacci, F. (eds.) iTrust 2006. LNCS, vol. 3986, pp. 179–192. Springer, Heidelberg (2006) 8. Lesani, M., Montazeri, N.: Fuzzy Trust Aggregation and Personalized Trust Inference in Virtual Social Networks. Computational Intelligence 25(2), 51–83 (2009) 9. Massa, P., Avesani, P.: Trust-aware Collaborative Filtering for Recommender Systems. In: Meersman, R. (ed.) OTM 2004. LNCS, vol. 3290, pp. 492–508. Springer, Heidelberg (2004) 10. Shekarpour, S., Katebi, S.D.: Modeling and Evaluation of Trust with an Extension in Semantic Web. Web Semantics: Science, Services and Agents on the World Wide Web 8, 26–36 (2010) 11. Victor, P., Cornelis, C., De Cock, M., Da Silva, P.P.: Gradual Trust and Distrust in Recommender systems. Fuzzy Sets and Systems 160(10), 1367–1382 (2008) 12. Victor, P., Cornelis, C., De Cock, M., Da Silva, P.P.: Practical Aggregation Operators for Gradual Trust and Distrust. Fuzzy Sets and Systems (article in press, corrected proof, 2011) 13. Zadeh, L.A.: Fuzzy Sets. Information Control 8, 338–353 (1965)

A Function Based Fuzzy Controller for VSC-HVDC System to Enhance Transient Stability of AC/DC Power System Niranjan Nayak, Sangram Kesari Routray, and Pravat Kumar Rout Multidisciplinary Research Centre, S.O.A University Bhubaneswar, India [email protected], {routraysk,niranjannayak.el.kec}@gmail.com,

Abstract. This paper presents a robust function based optimized fuzzy controller for VSC-HVDC transmission link operating in parallel with an AC transmission line connecting a synchronous generator to infinite bus. A two input one output methodology based optimized non-linear fuzzy logic controller has been proposed to provide voltage support by means of reactive control at both ends; to damp power oscillations and improve transient stability by controlling either active or reactive power; and to control the power flow through the HVDC link. An improved Teaching Learning Algorithm is applied to tune the gains of the fuzzy controller. Comprehensive computer simulations are carried out to verify the proposed control scheme under several system disturbances like changes in short-circuit ratio, converter parametric changes, torque change and faults on the converter and inverter buses. Based upon the time domain simulations in MATLAB/SIMULINK environment, the proposed controller is tested and its better performance is shown compare with the conventional PI controllers with respect to voltage stability, damping of power oscillations and improving of transient stability.

1

Introduction

It becomes a challenge to increase power delivery by improving transfer capability with AC expansion options in meshed, heavily loaded ac networks satisfying the technical requirements such as dynamic performances. Fast control of active and reactive power of VSC-HVDC system can improve power grid dynamic performance under disturbances. In case of serious disturbance threatens system transient stability, fast power run-back and even instant power reversal control functions can be used to help maintain synchronized power grid operation[1-3]. VSC-HVDC system can also provide effective damping to mitigate electromechanical oscillations by active and reactive power modulation. A range of advanced control functions can be implemented in VSC-HVDC systems for enhancement of ac network steady-state and dynamic performance. An attempt has been made in this paper in the above mentioned prospective. VSC-HVDC can be operated in three modes: i) Constant DC voltage control mode ii) Constant active power control mode iii) Constant AC voltage control mode[4]. The B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 441–451, 2011. © Springer-Verlag Berlin Heidelberg 2011

442

N. Nayak, S.K. Rouray, and P.K. Rout

VSC-HVDC transmission link has four control inputs, namely modulation signal magnitude and phase angle at both the converter stations and their interaction makes system a truly non-linear multi input and multi output control system. The control of all the possibilities is normally achieved through proportional-plus-integral (PI) controllers. These PI controllers suffer from the inadequacies of providing suitable control and transient stability enhancement over a wide range of system operating conditions under various faulted situation. Several authors have presented mathematical models and control strategies for VSC-HVDC transmission that include small signal stability [5], decoupling control in converter stations using feedback linearization [6], LMI based robust control [7], and adaptive control [8-10]. Most of these papers have considered a point to point VSCHVDC system without considering a parallel AC line in which case the sending and receiving ends are completely decoupled. Although some research has been conducted for parallel AC-VSC-HVDC systems [11-14], to explore the possibility of their smooth operation, till a lot of aspects are yet to be examined thoroughly. However also these previous studies were at the expense of neglecting a number of very important features in HVDC system operation, particularly related to the HVDCHVAC interactions. In this study, a robust nonlinear control is applied to the HVDC Light transmission systems in order to improve the dynamic behavior performance under wide range of operating and faulted conditions. The controller design is based on a simple function based optimized Fuzzy Controller. Here the gains of the Fuzzy Controller are optimized using a modified Teaching Learning Technique. This paper clearly describes the advantages of function based fuzzy controller over conventional PI Controller. The effectiveness of the robust controllers is demonstrated using the simulation studies with the aid of the MATLAB software package. The simulation results show that the controllers contribute significantly to improve the dynamic behavior of the HVDC light transmission system under a wide range of operating conditions, enhance the system stability and damp the undesired oscillations. The rest of the paper is organized in six sections as follows. In section 2, the time domain mathematical modeling of the VSC-HVDC light transmission system is presented. In section 3, the conventional PI controller and two rules based Fuzzy Logic Controller is briefly explained. The improved Teaching Learning Algorithm is explained in detail in Section 4. Simulation study that illustrates the effectiveness of the proposed strategies is discussed in section 5. At last, conclusions are drawn in Section 6.

2

Mathematical Modeling in D-Q Frame of Reference

In this paper mathematical model of the whole system has been derived in d-q frame. For the simplicity of analysis the three phases of the AC system are assumed to be symmetrical and the voltage of the AC source is in the q-axis exactly. Thus the reactive power can be controlled by the d-axis component and the active power or can be controlled by q axis component.

A Function Based Fuzzy Controller for VSC-HVDC System

443

 

UW

9V

9U

 

    

[W

UW /RDG

U

[W



 9&& 

[U

9 

9L

 &

[L

 

$& )LOWHU

U

$& )LOWHU

U

[

/RDG





Fig. 1.

The whole model is based upon the state space control theory ( x = A . x + B .u ). The controllers have been designed according to the system state. The model studied in this paper is shown above in Fig.1. The VSC HVDC system shown in fig.1 has a synchronous generator equipped with a voltage stabilizer and a parallel AC-HVDC Light system. xt1 , xt 2 , xt 3 and xt 4 are , are the reactance of the the transformer reactance. x1 is the line reactance and rectifier and inverter respectively. PL 1 , Q L 1 And P L 2 , Q L 2 are the loads in the passive network. DC line resistance is taken as the Rdc . C 1 and C 2 Are the DC link capacitors. V 1 And V R are the node bus and rectifier bus voltage respectively on the converter side network. V I and V 2 are the inverter bus and infinite bus voltage respectively on the inverter side network. 2.1

Synchronous Generator

The synchronous generator is described by a third order nonlinear mathematical model given by Ref. [15-16] δ = ω 1 − ω 0

ω =

Pm − Pe IC

(1)

1 {E q′ 0 + E ′f − E d′ − ( x ′d − x ′dd ). I 1d } E q′ = T0′

(2)

1 {− E ′f + k e (V1ref − V E ′f = T e′

(3)

)}

Where δ , ω , E ′f , E q′ are the power angle, angular speed, field voltage and quadrature axis voltage of the generator respectively. The excitation system of the generator consists of a simple AVR along with the supplementary PSS.

444

2.2

N. Nayak, S.K. Rouray, and P.K. Rout

Rectifire Station

The dynamic equation of converter station are given by V − V rD − Rr I rD IrD = 1 D + ω .I rQ + Lr Lr

V1Q − V rQ − Ri IrQ = − ω .I rD + I iQ Lr Li

− V d c 2  V 1  Pd c 1 ) Vd c 1 = − ( d c1  C  V d c1 rd c  

=

V 1 D I r D + V 1Q I r Q C d cV d c

−

I dc C dc

(4)

(5)

Where , and , are the direct and quadrature axis voltage of the ac bus , are the d-q axes current flowing into the and the rectifier bus respectively. rectifier. is the dc current flowing out of the rectifier. L r and C dc are the equivalent inductance offered by the rectifier and dc link capacitor respectively. The modulus index and firing angle can be related as

V rD = m 1 .V dc 1 . sin α 1

V rQ = m 1 .V dc 1 . cos α 1

(6)

The active and the reactive powers at the converter and the inverter sides are given by Pr = V rd I rd + V rq I rq Q r = − V rd I rq + V rq I rd

2.3

(7)

Inverter Station

The dynamic equations of the inverter are as follows V iQ − V 2 Q V − V2D are the d-q I iQ = − ω . I iD Where , + ω . I iQ , IiD = iD Li Li axes current flowing out of the inverter? L i is the equivalent inductance offered by the inverter. The modulus index

and firing angle

V iD = m 2 .V dc 2 . sin α 2

can be related as

ViQ = m2 .Vdc 2 . cos α 2

(8)

The active and the reactive powers at the converter and the inverter sides are given by

Pi = V id I id + V iq I iq

3

Q i = −V id I iq + V iq I id

(9)

Control Scheme for VSC-HVDC System

Since a parallel AC line is connected to a VSC-HVDC system both the rectifier and inverter stations are coupled giving rise to a system instability for certain operating conditions. Also due to coupling both of the sides are no more independent that is fault on one side reflects to the other. Normally PI controller is used for this purpose for providing damping to the system oscillations. Thus for improving damping

A Function Based Fuzzy Controller for VSC-HVDC System

445

performance significantly a novel two rule based fuzzy controller has been proposed that gives better stability to the system under abnormal conditions. 3.1

PI Controller

The PI controller is designed to provide decoupled and independent control of the DC link voltage and the generator side reactive power for converter station and the control of active power and reactive power for inverter station. The state variables of rectifier station are taken as Ird, Irq and Vdc1. For convenience dc voltage and reactive power of the rectifier are taken as the output variables. Since the reference bus d-axis voltage is taken as zero, the reactive power directly controlled by Irq. The state variables of inverter station are Iid, Iiq. Since the inverter station is operating in the active and reactive control mode the output states are taken as Iid, Iiq. The whole PI controller strategies for rectifier station and inverter station are shown in the Fig.2 and Fig. 3 respectively.  , L'UHI    ,L4UHI   

X

N S 

ǻ

,L' 

N S 

,L4

X

NU

NU V

X



 

X



P

ǻX

Fig. 2. PI controller for the rectifier station  ,L'UHI

B

   ,L4UHI    

N S 

N U V

,L'  B ,L4

N S 

N U  V

X X ǻX X X





 



ǻX

Fig. 3. PI controller for the inverter station

Here the trial and error technique is avoided and TLBO technique is used to optimize the values of the gains , , , , , . Both of the result has been demonstrated by simulation. 3.2

Two-Rule Based PI Like Fuzzy Controller

To overcome the tuning problem of conventional PI controller, here a two rule based PI like fuzzy controller has been proposed as shown in fig.4. This section describes the main features of the proposed two rule base two input one output FLC.

446

N. Nayak, S.K. Rouray, and P.K. Rout

The proposed FLC uses two inputs as error e, and change of error de, and one output du (change in control). PI like FLC is based upon two rules and one metarule as follows: R1: If e is EP and de is CEP then du is UP.,R2: If e is EN and de is CEN then du is UN. In the above rule base, the fuzzy sets for error e, change of error de, and change of control du are EP(error positive), and EN(error negative), CEP(change of error positive), and CEN(change in error negative), UP(control positive),and UN(control negative), respectively. Since the above rule base is not complete as at least four rules are required for the fuzzy PI or PD controller. Thus, a third metarule of the form given below is used to obtain the fuzzy 2-rule based controller:

“If e is not self correcting, then control action du is not zero and depends upon the sign and magnitude of e and de”. This is achieved by defining the relationship as sign (du) = sign (e). Thus the metarule takes the form “If sign (e) = sign (de) ≠ 0, Then sign (du) = sign (e)”.  5HI 'HOD\

)X]]\ /RJLF &RQWUROOHU

8

0HDVXUHG9DOXH Fig. 4. Generalized Fuzzy like PI controller

The membership functions for the fuzzy sets EP and EN are chosen as (10) μ1 = μEP(e) = .5(1+ tan(h(K1e)) ,( μEN (e) = 1 − μ1 = .5(1− tan(h(K1e)) Similar expressions hold for CEP, and CEN, respectively as (11) μ2 = μCEP(de) = .5(1 + tan(h(K2 de)) , μCEN(de) =1− μ2 = .5(1− tan(h(K2de)) The defuzzified output becomes du = (2/3) G ( ( A1 − A 2 ) /( A1 + A 2 ) where A1 = 0 . 5 λμ 1 + 0 . 5 (1 − λ ) μ 2 , ,

A2 = 0.5λ (1 − μ1 ) + 0.5(1 − λ )(1 − μ 2 )

(12)

Using the above relations and after some simplifications, the defuzzified control output is obtained as du = (2/3) G ( ( λ tanh( K 1 e ) + (1 − λ ) tanh( K 2 de )) (13) A slightly different version is obtained by using a product-t norm for the defuzzification process in which the value of A 1 becomes equal to A1 = A 2 exp ( K 1 e + K 2 de ), and using simplification du is obtained as du = (2/3) G tanh( K 1 e + K 2 de )

(14)

A Function Based Fuzzy Controller for VSC-HVDC System

447

The various gains used in the above equations like K 1 , K 2 , G are to be chosen appropriately for optimal control performance. Thus the control input at time t is u ( t ) = u ( t − 1) +

2 G tanh( K 1 e ( t ) + K 2 de ( t )) 3

(15)

Where K1 acts as integral constant and K2 as proportional constant. Keeping K1=0, G and K2 are tuned. Then K1 is gradually increased to achieve the required output. In this way all the gains are tuned at one operating condition but for different operating conditions the gains are optimized using the improved teaching learning algorithm. The converter and inverter station auxiliary fuzzy controllers can be written by using equation (30) separately as follows. For the converter station 2 2 , u f 1 (t ) = u1 f 1 (t − 1) + G1 tanh(K11e1 (t ) + K12 de1 (t )) u f 2 (t) = u f 2 (t −1) + G2 tanh(K21e2 (t) + K22 de2 (t)) 3 3

(16)

Where e1 = I ref' − I ' , de1 = e1 (t ) − e1 (t −1) rD rD e 2 = V dcref1' − V dc 1' , de 2 = e 2 ( t ) − e 2 ( t − 1)

(17)

For inverter station 2 u f 3 (t ) = u f 33 (t − 1) + G3 tanh(K 31e3 (t ) + K 32 de3 (t )) 3 u f 4 ( t ) = u f 4 ( t − 1) +

2 G 4 tanh( K 41 e 4 (t ) + K 42 de 4 (t )) 3

Where e 3 = I iDref ' − I iD ' , de 3 = e3 (t ) − e 3 (t − 1) And e 4 = I

4

ref iQ

''

− I iQ ' , de 4 = e 4 ( t ) − e 4 (t − 1)

(18) (19) (20)

Teaching-Learning Based Optimization (TLBO)

This is a population based method based on the effect of influence of a teacher on learners. Due to faster convergence rate, better searching capability to find global optimum, less adjustment of algorithm parameter change and less computational effort than the other nature inspired algorithm it is used to tune the fuzzy controller parameters. Teacher Phase: In this phase learning of the learner from the teacher is encapsulated by adding a factor to the old value as follows. The updated values , in the iteration are calculated by adding some percentage of the difference between the teacher and the individual learner subject value to the old value , as follows. Xnew, s = Xold,s + r and(Xteacher,s – Tf(Meanind-sub,s))

(21)

Learning Phase: In this phase the second way of learning of the learner by interaction with other learners is taken care of by adding a factor to the above value , .as

448

N. Nayak, S.K. Rouray, and P.K. Rout

follows. But with a condition a learner learns new if the other learner has more knowledge comparatively. ,

,

if

(22)

,

,

if

(23)

Stopping Criterion: The process terminate once the maximum generation number is achieved. To validate the established steady state model and the proposed control strategy, the operating points are taken into consideration are mentioned in appendixA for theoretical simulation studies in MATLAB. The fitness function chosen here for finding optimized PI and Fuzzy controller gains is chosen as 1 1 ∑ 1

5

Simulation Result

Case-1. The system is simulated with a LLLG fault at bus-1 from 1 sec to 1.2 second. Due to the fault the ac voltage at the corresponding bus is decreased to a critical level. The performance of the optimized fuzzy controller restores the system faster than the conventional PI controller. The performance improvement with the new controller shown in Fig.5 clearly indicates the robustness of the proposed controller. The damping property marginally improves along with the reduction in overshoot and settling time. 1

0.88

PI OF

0.6

0.84 0.82 0.8 0.78 0.76 0

0.4 0.2 0 -0.2

1

2

3

4

5

-0.4 0

6

time (second)

1

2

3

4

5

6

time (second)

Fig. 5. (a) Variation of power angle 0.88

Fig. 5. (b) Variation of rotor speed 0.9

PI OF

0.86

PI OF

0.8 0.7

Qr (Pu)

0.84

Pr (Pu)

PI OF

0.8

Omega (Pu)

Delta (Pu)

0.86

0.82

0.6 0.5

0.8 0.4

0.78 0.76 0

0.3

1

2

3

4

5

time(second)

Fig. 5. (c) Variation of rectifier power

6

0.2 0

1

2

3

4

5

time (second)

Fig. 5. (d) Variation of Qr at rectifier

6

A Function Based Fuzzy Controller for VSC-HVDC System

449

Case-2. With operating condition of Pr=0.8Pu andQr=0.6Pu,the 10-cycle LLLG fault is simulated on the inverter side. Fig.6 shows the satisfactory performance of the proposed optimized Fuzzy controller in comparison to the optimized PI controller. The proposed controller maintains a complete coupling effect between the stations. The ac side of the rectifier station operates completely in a stable condition and is very less affected by the fault on the inverter side. Fig. 6 indicates 1

3

PI OF

Omega (Pu)

0.8

Delta (Pu)

PI OF

2

0.6

1 0

0.4

-1 0.2 0

1

2

3

4

5

6

-2 0

time (second)

3

4

5

6

Fig. 6. (b) Variation of rotor speed

1.4

PI OF

1.2

1.5

PI OF

1

1 0.8

Qi (Pu)

Pi (Pu)

2

time (second)

Fig. 6. (a) Variation of power angle

0.6 0.4 0.2 0 0

1

0.5 0 -0.5

1

2

3

4

5

6

-1 0

time (second)

1

2

3

4

5

6

time ( second)

Fig. 6. (c) Variation of Pi at inverter

Fig. 6. (d) Variation of Qi after the rectifier

Case-3. Here a 1 cycle fault at bus 1 is created by increasing the mechanical input to the generator (20 percent of the preliminary value) with an operating point Pr=1.03Pu, Qr=0.3pu, is simulated on the inverter side. Fig.7 shows the satisfactory improvement of the power system by the proposed controller. 0.1806

0.4

OF PI

0.1804

0.3 Omega (Pu)

delta (Pu)

0.1802 0.18 0.1798 0.1796

0.25 0.2 0.15

0.1794 0.1792 0

PI OF

0.35

0.1

1

2

3

4

5

time (second)

Fig. 7. (a) Variation of power angle

6

0.05 0

2

4

6

8

10

time (second)

Fig. 7. (b) Variation of rotor speed

12

450

6

N. Nayak, S.K. Rouray, and P.K. Rout

Conclusion

The VSC based HVDC light system with optimized function based fuzzy controller has been proposed and designed in this paper. The gains of the proposed controller have been tuned using a adaptive Teaching-Learning based optimization technique. The methodology used to design the PI like FLC for the HVDC light system works effectively. Under normal and fault condition the controller is tested which gives the satisfactory result. Simulation result shows that with proposed control strategy, quick response and dynamic stability have been achieved for any kind of changes and high level control accuracy is attained at different operating conditions of the system as well as for a variety of disturbances occurring in the network.

References [1] Andersen, B.R., Xu, L., Horton, P.J., Cartwright, P.: Topologies for VSC Transmission. Power Engg. Journal 16, 142–150 (2002) [2] Asplund, G.: Application of HVDC Light to power system enhancement. IEEE Power Eng. Soc. Winter Mtg. 4, 2498–2503 (2000) [3] Zhang, G., Xu, Z.: Steady-state model for VSC based HVDC system and its controller design. IEEE Power Engineering Society Winter Meeting 3, 1085–1090 (2001) [4] Durrant, M., Werner, H., Abbott, K.: Synthesis of multi-objective controllers for a VSC HVDC terminal using LMIs. In: IEEE Conference on Decision and Control, pp. 4473– 4478 (December 2004) [5] Ruan, S.Y., Li, G.J., Jiao, X.H., Sun, Y.Z., Lie, T.T.: Adaptive control design for VSCHVDC system based on backstepping approach. Electric Power System Research 77(56), 559–565 (2007) [6] Ruan, S.X., Li, G.J., Peng, L., Sum, Y.Z., Lie, T.T.: A non-linear control for enhancing HVDC Light transmission system stability. Electrical Power Energy Systems 29, 565– 570 (2007) [7] Xu, L., Anderson, B.R., Cartorite, P.: VSC transmission operating under unbalanced AC conditions- analysis and control design. IEEE Transactions Power Delivery 20(1), 427– 434 (2005) [8] de la Villa Jean, A., Acha, E., Exposito, A.G.: Voltage source converter modeling for power system state estimation: STATCOM and VSC-HVDC. IEEE Transactions Power System 23(4), 1552–1559 (2008) [9] Latorre, H.F., Ghandhari, M., Soder, L.: Control of a VSC-HVDC Operating in Parallel with AC Transmission Line. In: IEEE Conference (2006) [10] Li, C., Li, S., Han, F., Shang, J., Qiao, E.: A VSC-HVDC fuzzy controller to damp oscillation of AC/DC Power System. In: IEEE Conference, ICSET 2008, pp. 816–819 (2008) [11] Latorre, H.F., Ghandhari, M., Soder, L.: Active and Reactive Power Control of VSCHVDC. Electrical Power System Research 78, 1756–1763 (2008) [12] Scholtz, E.: Towards realization of a highly controllable transmission system-HVDC light. ABB Corporate Research, 1–21 (2008)

A Function Based Fuzzy Controller for VSC-HVDC System

451

Appendix Synchronous machine data Generator rating = 900 MVA,Base MVA = 500

Pe = 1.4 pu, Qe =0.8 pu,,Pt = 0.5 pu,Qt = 0.2 pu,Pdc =0.3 pu Load active power PL =0.7 puLoad reactive power, QL =0.3 Pu = 30,

1.8,

0.3,

1.7, H = 6.5, τ’d0=8

PSS data U pss

ST q = K δ . 1 + ST

 1 + ST 1    1 + ST   2 

K

. q

0.138;

= 0 . 24 , T q = 0 . 4 , T 1 = 0 . 03 , T 2 = 0 . 01

,

= 1.98; Optimized PI gains .656; = 0.39; = -1.57; Optimized fuzzy control gains

δ

= 3.56;

K

11

= 0.341;

K 22 = 2.49; G 2 = 0.13; K 31 = 0.412; =0.237; K 42 = 3.92; G 4 = 0.19;

= 0.45;

K

12

= -1.3;

= 4.593;

= .175;

G

1

=

= 0.37; K 21 =

K 32 = - .593; G 3 = 0.63; K 41

Control parameters of TLBO

Number of particles = 20; Number of variables for optimized fuzzy controller = 12; The maximum and minimum limits for the gains: 10 and -5.

A Bayesian Network Riverine Model Study Steven Spansel, Louise Perkins, Sumanth Yenduri, and David Holt University of Southern Mississippi, 730 East Beach Boulevard, Long Beach, MS 39560

Abstract. In this paper we apply a Bayesian Network Methodology to a riverine model study. We show that continuous variables, following Ebert-Uphoff’s method, would improve this model, and we modify an existing share-ware Bayesian Network to simulate continuous variables. We also present a new metric for the Bayesian Network.

1

Introduction

Bayesian Networks are software tools that encode causal relationships. Graphically they are represented as Directed Acyclic Graphs (DAG’s). Each DAG node represents a random variable that can take on two or more discrete values. Each DAG arc represents a causal conditional dependent relationship between the two nodes; a non-zero covariance relationship is represented only if no other arc implies the same dependence. Mathematically, two random variables, A and B, are conditionally independent, given C, if any of the conditional probability equations below hold: , | | , | ,

|

| |

(1) (2)

|

(3)

If we consider the set of discrete values that our stochastic variables can assume as bins, the set of bins must of necessity cover the full range of the continuous variable being discretized. In general these bins form a partition over the values within the range; this is a hard discretization. Bayesian Networks propagate new probability tables for nodes via Bayesian inference, which is based on Bayes’ Theorem. For a case where a node representing variable A has n bins, the general form of Bayes’ Theorem can be used to find the probability of some , given new evidence E [1]. A third, "chained" version of the theorem can be used to find the probability that some event C has occurred, given that events A and B have occurred [2]. In a multi-connected network calculations would need to be performed, each themselves requiring linear time, bringing the total computation complexity cost to quadratic time complexity [3]. To speed up this process, Bayesian Network software packages use join tree algorithms [4] to decompose the graph represented by the Bayesian Network into a join tree. After this, a propagation algorithm is executed on the join B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 452–459, 2011. © Springer-Verlag Berlin Heidelberg 2011

A Bayesian Network Riverine Model Study

453

tree. There are many established algorithms [5] for performing this propagation. Many general use, pre-built implementations are available from a variety of sources. For example, there are software development firms that produce high quality commercial products. In addition, several University Computer Science / Mathematics Departments maintain free (with license limitations), open-source implementations. There are two primary characteristics that separate these implementations. The first is the method of use. Many implementations are only available for certain Operating Systems (most commonly, Microsoft Windows). Some implementations can only be used via a graphical user interface (GUI). Such implementations cannot be automated by incorporating them into a computer program. Still other implementations are not wide enough in their scope. For instance, there are some that focus exclusively on using Bayesian Networks to perform sensitivity analysis. The second distinguishing characteristic among Bayesian Network software packages is their implementation of the junction tree algorithm. In order to be deemed correct, junction tree algorithms must arrive at the correct answer for all computations. However, some implementations of this algorithm perform these calculations much more efficiently than other implementations. This is one of the main areas of competition between competing commercial Bayesian Network software implementations. For a list of some available packages, see Appendix 2 of Agena Risk’s Knowledge Transfer Report on Bayesian Network technology [6]. We utilize the Netica Bayesian Network software package, produced by [7]. It is available for all major platforms (Microsoft Windows, Macintosh OS X, Linux, and Sun Solaris). Also, it can be utilized both via a GUI and via Application Programming Interfaces (APIs) available for many different programming languages, including Java, C, C++, .net, and others. Furthermore, it has earned a reputation as being a fast and stable implementation through its performance in other scientific research projects. We use Netica in two forms: as a Java API and as a Windows GUI. Versions of this software can be downloaded from http://www.norsys.com/netica-j.html and http://www.norsys.com/download.html, respectively. Once downloaded, no installation of the software is necessary. The versions that can be freely downloaded but are limited in size – the number of nodes that can be contained by a network is limited. Furthermore, there are legal restrictions on the use of the software. These can be viewed online at http://www.norsys.com/netica-j.html.

2

Related Research

[8] propose a probability based soft discretization. We present a solution that mimics a soft discretization and works with existing Software Tools. Hard discretization of a variable loses accuracy. Precisely, if a value of x is allocated to a bin with range [x-.5] – [x+.2], the effect of x changes to that of x-1.5 – the midpoint of the bin. Commonly available Bayesian Network software packages only allow hard discretization of continuous variables [8]. They propose a probability-based soft discretization solution to this problem. Soft discretization is a weighted scheme in which a value does not have to belong exclusively to one bin. For instance, the value 18 would be given a large weight in bin 2, and a smaller weight in bin 3. In this way, less

454

S. Spansel et al.

of the information about where a value lies within the bin is lost. Modifying the Bayesian Model was beyond the scope of this research work. Instead we simulated soft discretization by relaxing our metric a posterori. The use of the junction tree algorithm for efficient probability propagation in multi-connected networks was first described by Lauritzen & Spiegelhalter [4]. There are now, however, many different algorithms for this purpose, as described by Neapolitan [5]. All of these algorithms begin by moralizing the network, making it chordal, and decomposing it into a junction tree. After that, the algorithms vary as to how they actually perform the propagation of probabilities throughout the network. See, for example, Jensen & Jensen [9], Weisstein [10], and McGlohon [11] to gain a better understanding of this process.

3

Data Location and Sobek Model

The United States Geological Survey (USGS) maintains data collection stations at multiple sites along the Kootenai River, located in northern Idaho. We utilize a numerical model constructed to replicate the river flow properties at the "Tribal Hatchery" site, located approximately 150 miles upstream from the mouth of the river. The Sobek model is one dimensional, and includes momentum and continuity equations for a variety of flow dynamics including sediment load. It has been verified to be in the regions of strong sediment transfer [12]. It uses the Boussinesq approximation and includes hydraulic parameterization, as well as the standard cross sectional area flow parameters. A typical river model might require several years to construct, and decades to calibrate. For this reason, we utilize a set of results from an existing, validated numerical riverine model of the Tribal Hatchery site. This model data was generated by the SOBEK model suite, produced by Delft Hydraulics [13]. This powerful software tool simulates one dimensional integral processes in a water drainage system – river, canal, etc [12]. The model was configured specifically for the Tribal Hatchery site mentioned above, and was run over a range of expected conditions. This produced a table of 87,840 cases, each consisting of a value for Discharge, Velocity, Width, and Depth. We simulate having set up and run the model ourselves by selecting cases from this pre-run result set. Test case selection was performed using a well seeded pseudo random number generator. This was selected because the region exhibits significantly different flow regimes episodically and we wanted to capture a wide variety of flows. One of the earliest studies of river dynamics was based on a study of the Po river made by the Italian engineer Guglielmini in the seventeenth century. These early studies used a transport equation with simply related variables, as Q= V* A (where A is the cross sectional area, V is velocity and Q is the discharge) . Such simple equations assume idealized flow. For example, it assumes conservation of water. Yet water may be absorbed by dry ground when a river’s depth increases and a dry river bank is first wetted.

A Bayesian Network Riverine Model Study

4

455

Performance Metric: Bin Prediction Accuracy

To test the network, we give the network values for the other 3 variables (not for the variable being tested) for all 87,840 cases. We compare the bin which the network estimates the value to be in (for each case) with the bin the value was actually in. We use a modified Bin Prediction Accuracy metric as our performance criteria – the percentage of times the network accurately chooses the correct bin. It is desirable that the matrix from which BPA is calculated has high values along the principal diagonal, indicating that the Bayesian Network predicted the correct bin. In a "strict" BPA calculation, only the values along that principal diagonal would be counted as correct. However, Ebert-Uphoff [8] discusses the fact that, in a hard discretization scheme (such as ours), intra-bin information is lost. Thus, for values that fall near the edge of a bin, there is an increased likelihood that the Bayesian Network will predict the bin next to the “correct” bin. Figure 1 clearly shows that there are many bins with high prediction counts that are next to the actual bins the values fall into. If we only counted the actual “correct” bins as correct, we would be ignoring the distributional information inherent in the matrix – we would be counting those cases the same as when the network’s prediction was completely wrong. Thus, we include those bins as "correct" in our BPA calculations. This soft correction is an a posteriori accommodation, in contrast to Ebert-Uphoff’s [8] a priori soft discretization, intended to accomplish the same goal, albeit less rigorously.

Fig. 1. Bin Prediction Accuracy: Predicted Bin Vs Actual Bin

5

Discretization Optimization

Discretization involves answering two questions: Where should the bin boundaries (’breaks’) be placed? How many bins should there be? Optimal solutions to these

456

S. Spansel et al.

questions are computationally intensive and often not intuitive. In Netica (and other Bayesian Network software packages), the default method of answering the first question is to use a quantile method to place the breaks such that an even number of cases falls into each bin. There is no default answer to the second question; researchers generally must explicitly choose the number of bins. In this paper, we develop an algorithm to answer both of these questions in a way that improves the overall accuracy of the resulting network. In the quantile method, bin boundaries are placed such that an even number of cases falls into each bin. Essentially, it orders the cases, then splits them into evenly spaced groups. For instance, if we were using 20 bins, the first 5% would go into bin 1, the next 5% into bin 2, and so on. It seems intuitive that placing bins such that an even number of cases fall into each bin would yield high BPA. The problem, however, lies in the fact that the bin boundaries are placed at pre-determined places in the Cumulative Distribution Function. So bin boundaries can (and often do) get placed in the middle of a dense cluster of cases that would preferably be grouped together in one bin because they represent similar dynamics. A data clustering approach addresses this bin boundary placement directly. This method locates these dense clusters of cases, and places the bin boundaries such that the clusters are binned together, forming natural breaks. As evidenced in table 1, the optimal bin count combination using this method yields approximately 8% higher Bin Prediction Accuracies than the optimal bin count combination using the quantile method. Table 1. Discretization Optimization Results

For verification testing, we use a brute-force approach to determine the optimal number of bins for each variable. We test all even numbers between 6 and 20 as bin counts for each variable, using both methods (quantile and data clustering) mentioned above. For each bin count combination, we record the BPA for each variable. For each bin boundary placement method, we choose the bin count combination that yields the highest mean BPA. Discretization optimization leads to increased prediction accuracy —-bins matter. But discretization also yields another benefit. Because it may lead to the use of fewer bins per variable, discretization optimization may yield faster computing times, since there are less bins across which the junction tree algorithm must propagate probabilities. Using all 87,840 cases from the SOBEK model

A Bayesian Network Riverine Model Study

457

output, the Bayesian Network is able to predict a single unknown variable with 98.1% accuracy, given values for the other three variables, as shown in table 1. Here, however, we show that the Bayesian Network can reach nearly the same prediction accuracy levels, using far fewer than the entirety of the 87,840 cases. In our study a random selection of only 1,000 cases resulted in accuracy above 90% (Fig 2). The network’s ability to accurately predict which bin a variable’s value will fall into (1) quickly reaches nearly the same accuracy levels as using all 87,840 cases, and (2) quickly shows diminishing returns – to add more data would be of decreasing benefit.

Fig. 2. Bin Prediction Accuracy Vs Number of Cases

6

Future Development

Our study showed that the network can achieve essentially the same BPA using far fewer than the full set of 87,840 cases. Thus, there should be an informed way to intelligently choose data points from the full set, rather than using random selection —an uninformed method. A consequence of discretizing the SOBEK model data is a loss of information. For instance, consider a situation in which bin 3 of variable V contains all values in the range 3 to 5. In this situation, the difference between a value of 3.1 and a value of 4.9 has been lost, since both fall into bin 3. It seems likely that the network’s BPA would benefit from the use of soft discretization [8]. This is a weighted discretization scheme in which it is not necessary to assign all of a value’s weight to only one bin. For instance, the value of 3.1 might have a large weight assigned to bin 3, and a smaller weight assigned to bin 2, indicating that it was near the lower bound of bin 3. In this way, we would save some of the information that is now being lost due to the hard discretization scheme. However, as [8] discusses, there are

458

S. Spansel et al.

currently no Bayesian Network software packages that allow soft discretization. Thus, a Bayesian Network software package would have to be modified to allow this.

7

Conclusion

As mentioned in the section describing BPA, we include both the bins next to and on the principal diagonal in Fig.1 as correct. It is likely that there is a more optimal way to include these bins in our calculations. Perhaps an exponentially decreasing weight would be more appropriate. In this scheme, the bins along the principal diagonal would be counted as correct with a weight of 1. Bins that are one bin away from the principal diagonal would be counted, but with a smaller weight, and bins that are two bins away would be counted, but with a still smaller weight. More work is needed to determine the coefficient weights for an exponentially decaying scheme. The discretization improvements to Bayesian network forecast methods are general in nature, and may be applied to many other numerical models. The work is not limited to our riverine model. Netica’s default discretization scheme is quantile, usually using the same number of bins for every variable. Using this scheme (with twenty bins for each of our four variables), the network achieved 90.1% accuracy. We developed a discretization optimization algorithm to select a discretization scheme that yields a network that is both more accurate and more computationally efficient. This algorithm uses data clustering to find natural breaks between groups of clustered data. By changing the discretization scheme and intelligently selecting the number of bins for each variable, we increased the network’s accuracy from 90.1% to 98.1%. It is important to note that the value of these contributions is not limited to the application of Bayesian Networks in a riverine system. Rather, the concepts are general enough that they can be applied to any system in which the data must be discretized before it can be used in a Bayesian Network. Thus, they should be of benefit not only to geoscientists, but also to physicists, engineers, economists, and researchers in any discipline to which numerical modeling can be applied. We modified our accuracy measure as well. It is an improvement over a strict bin prediction accuracy metric in that it maintains not only cases in which the network predicted the correct bin, but also cases in which the network was only off by one bin. In a stricter metric, these cases would be counted equally as wrong as ones for which the network predicted a completely wrong bin. This metric is also an improvement over the standard hindcasting method of determining a network’s accuracy. In hindcasting, the actual value is compared with the mean of the bin predicted by the network. So, for all cases in which the value does not fall exactly in the center of the bin, the network is penalized as having "error", even though (since it is discretized) it cannot perform any better than predicting the correct bin. In short, hindcasting attempts to extract a continuous value from a discrete variable. Our metric improves on this by only trying to extract a discrete value from a discrete variable. Because the network can do no better than predicting the correct bin, that is counted as completely correct in our modified bin prediction accuracy metric.

A Bayesian Network Riverine Model Study

459

Finally, we showed that the network’s accuracy exhibits diminishing returns as more data is added. This should be a significant help to researchers to help them determine how much data needs to be collected from laboratory and field experiments in order to achieve results that are at least within some pre-defined accuracy threshold. Further, the iterative learning algorithm we developed can be used (in conjunction with a numerical model) to determine, a priori, approximately how many data points must be collected to achieve a desired accuracy level. This can save researchers much valuable time, money, and resources that could otherwise be wasted on collecting unnecessary data.

References 1. 2. 3. 4.

5.

6.

7. 8. 9.

10. 11. 12. 13.

Agena Bayes Theorem, http://www.agenarisk.com/resources/probability_puzzles/ bayes_theorem.shtml Niedermayer, D.: An Introduction to Bayesian Networks and their Contemporary Applications (December 1998), http://www.niedermayer.ca/papers/bayesian/index.html.old Russell, S.J., Norvig, P.: Artificial Intelligence: A Modern Approach, 3rd edn. Pearson Education (2009) Lauritzen, S.L., Spiegelhalter, D.J.: Local Computations with Probabilities on Graphical Structures and their Application to Expert Systems. Journal of the Royal Statistical Society 50(2), 157–224 (1988) Neapolitan, R.E.: Learning Bayesian Networks. In: The 13th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD 2007, p. 5:1. ACM, New York (2007) Fenton, N., Neil, M.: Managing Risk in the Modern World: Applications of Bayesian Networks, Technical report, London Mathematical Society and Knowledge Transfer Network for Industrial Mathematics (2007) Norsys Software Corporation: Netica Bayesian Network Software (2011), http://www.norsys.com Ebert-Uphoff, I.: A Probability-Based Approach to Soft Discretization for Bayesian Networks, Technical report, Georgia Institute of Technology (2009) Jensen, F.V., Jensen, F.: Optimal Junction Trees. In: Proceedings of the Tenth Conference on Uncertainty in Artificial Intelligence, pp. 360–366. Morgan Kaufmann Publishers Inc., San Francisco (1994) Weisstein, E.W.: Chordal Graph, From MathWorld–A Wolfram Web Resource, http://mathworld.wolfram.com/ChordalGraph.html McGlohon, M.: Methods and Uses of Graph Demoralization, http://www.cs.cmu.edu/~mmcgloho/pubs/sigbovik3.pdf Ji, Z., de Vriend, H., Hu, C.: Application of Sobek Model in the Yellow River Estuary. In: International Conference on Estuaries and Coasts, Hangzhou, China (November 2003) Delft Hydraulics Software: SOBEK Suite (2011), http://delftsoftware.wldelft.nl/index.php?task=blogcategory& Itemid=35

Application of General Type-2 Fuzzy Set in Emotion Recognition from Facial Expression Anisha Halder1, Rajshree Mandal2, Aruna Chakraborty2, Amit Konar1, and Ramadoss Janarthanan3 1

Department of Electronics and Tele-Communication Engineering, Jadavpur University, Kolkata-32, India 2 Department of Computer Science and Engineering, St. Thomas’ College of Engineering and Technology, Kolkata, India 3 Department of IT, Jaya Engg. College, Chennai {halder.anisha,rajshree.mondal}@gmail.com, [email protected], [email protected], [email protected] Abstract. This paper proposes a new technique for emotion recognition of an unknown subject using General Type-2 Fuzzy sets (GT2FS). The proposed technique includes two steps- first, a type-2 fuzzy face-space is created with the background knowledge of facial features of different subjects containing different emotions. Second, the emotion of an unknown facial expression is determined based on the consensus of the measured facial features with the fuzzy face-space. The GT2FS has been used here to model the fuzzy face space. The general type-2 fuzzy involves both primary and secondary membership distributions which have been obtained here by formulating and solving an optimization problem. The optimization problem here attempts to minimize the difference between two decoded signals: the first one being the type-1 defuzzification of the average primary membership distributions obtained from nsubjects, while the second one refers to the type-2 defuzzified signal for a given primary distribution with secondary memberships as unknown. The uncertainty management policy adopted using general type-2 fuzzy set has resulted in a classification accuracy of 96.67%. Keywords: Emotion Recognition, Facial feature extraction, Type-2 primary membership, Type-2 secondary membership, Fuzzy Face Space.

1

Introduction

Emotion recognition is currently gaining importance for its increasing scope of applications in human-computer interactive systems. Several modalities of emotion recognition, including facial expression, voice, gesture and posture have been studied in the literature. However, irrespective of the modality, emotion recognition comprises two fundamental steps involving feature extraction and classification. Feature extraction refers to determining a set of features/attributes, preferably independent, which together represents a given emotional expression. Classification aims at mapping emotional features into one of several emotion classes. B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 460–468, 2011. © Springer-Verlag Berlin Heidelberg 2011

Application of General Type-2 Fuzzy Set in Emotion Recognition

461

Among the well-known methods of determining human emotions, Fourier descriptor [1], template matching [2], and neural network techniques [3], [4] deserve special mention. Other important works undertaken so far for recognition of emotions from facial expression by selecting suitable features include [5], [6], [7] and by identifying the right classifier include [3], [4],[10]. The paper provides an alternative approach to emotion recognition from an unknown facial expression, when the emotion class of individual facial expression of a large number of experimental subjects is available. The General Type-2 fuzzy set (GT2FS) based approach employs to construct a fuzzy face space, comprising both primary and secondary membership distributions, obtained from known facial expressions of several subjects containing multiple instances of the same emotion for each subject. The emotion class of an unknown facial expression is determined by obtaining maximum support of each class to the given facial expression. The class with the maximum support is the winner. The maximum support evaluation here employs both primary and secondary distributions. Experiments reveal that the classification accuracy of emotion of an unknown person by the GT2FS based scheme is as high as 96%. The paper is divided into five sections. In section 2, we propose the principle of GT2FS approach. Methodology of the proposed scheme is discussed in section 3. Experimental details are given in section 4. Conclusions are listed in section 5.

2

Principles Used in the GT2FS Approach

The GT2FS based reasoning realized with measurements taken from n-subjects, requires k × m × n general type-2 fuzzy sets to determine the emotion class of an unknown facial expression where, k is the number of emotion classes and m is the number of features. Let F={f1, f2, …,fm} be the set of m facial features. Let μ A~ ( fi ) be the primary ~ membership in [0,1] of the feature fi to be a member of set A , and μ ( fi , μ A~ ( fi )) be

the secondary membership of the measured variable fi in [0,1]. If the measurement of a facial feature, fi, is performed p times on the same subject experiencing the same emotion, and the measurements are quantized into q intervals of equal size, we can evaluate the frequency of occurrence of the measured variable fi in q quantized intervals. The interval containing the highest frequency of occurrence then can be identified, and its centre, mi, approximately represents the mode of the measurement variable fi. The second moment, σi, around mi is determined, and an exponential bellshaped (Gaussian) membership function centered on mi and with a spread σi is used to represent the membership function of the random variable fi. This function represents the membership of fi to be CLOSE-TO the central value, mi. It may be noted that a bell-shaped (Gaussian-like) membership curve would have a peak at the centre with a membership value one, indicating that membership at this point is the largest for an obvious reason of having the highest frequency of fi at the centre. On repetition of the above experiment for variable fi on n subjects, each experiencing the same emotion, we obtain n such membership functions, each one for

462

A. Halder et al.

one individual subject. Naturally, the measurement variable fi now has both intra- and inter-personal level uncertainty. The intra-level uncertainty occurs due to the preassumption of the bell-shape (Gaussian distribution) of the primary membership function, and the inter-level uncertainty occurs due to multiplicity of the membership distribution for n subjects. Thus a new measurement for an unknown facial expression can be encoded using all the n-membership curves, giving n possible membership values, thereby giving rise to uncertainty in the fuzzy space. GT2FS approach tunes the primary membership values for the given measurements using the secondary memberships of the same measurement, and thus reduces the degree of intra-level uncertainty of the primary distributions. The reduction in the degree of uncertainty helps in improving the classification accuracy of emotion at the cost of additional complexity required to evaluate type-2 secondary distributions and also to reason with k × m × n fuzzy sets. Let fi/ be the measurement of the i-th feature for an unknown subject. Now, by consulting the n primary membership distributions obtained from n-subjects for a given emotion, we obtain n primary membership values for the measurement fi/ given by μ1A ( fi / ), μ A2 ( f i / ),......., μ An ( fi / ) . Let the secondary membership values for each primary membership respectively be μ ( f i / , μ1A ( f i / )), μ ( fi / , μ A2 ( fi / )),......., μ ( f i / , μ An ( fi / )) . Since the secondary memberships denote the degree of accuracy of the primary memberships, the uncertainty in a primary membership distribution can be reduced by multiplying each primary membership value by its secondary membership. Thus the modified primary membership values are given by mod 1 μ A ( fi / ) mod 2 μ A ( fi / )

= μ1A ( f i / ) × μ ( f i / , μ1A ( f i / )),

= μ A2 ( f i / ) × μ ( f i / , μ A2 ( f i / )), … … … … … … … …. … … … mod n μ A ( f i / ) = μ An ( fi / ) × μ ( fi / , μ An ( fi / ))

where

mod

(1)

μ Aj ( fi / ) denotes the modified primary membership value for j-th subject. The

next step is to determine the range of

mod

μ Aj ( fi / ) for j= 1 to n, comprising the mini-

mum and the maximum given by [ mod μ ~ ( f i / ) , A mod

mod

μ A~ ( f i / )] , where

μ A~ ( f i / ) = Min { mod μ 1A~ ( f i / ), mod μ A2~ ( f i / ),..., mod μ A~n ( f i / )},

mod

μ A~ ( f i / ) = Max { mod μ 1A~ ( f i / ), mod μ A2~ ( f i / ),..., mod μ An~ ( f i / )},

If there exist m different facial features, then for each feature we would have such an interval, and consequently we obtain m such intervals given by [

mod

μ A~ ( f1/ ) ,

mod

μ A~ ( f1/ )] , [ mod μ A~ ( f 2/ ) ,

mod

μ A~ ( f 2/ )] ,... [ mod μ A~ ( f m/ ) ,

mod

μ A~ ( f m/ )]

The proposed reasoning system employs a particular format of rules, commonly used in fuzzy classification problems. Consider for instance a fuzzy rule, given by

Application of General Type-2 Fuzzy Set in Emotion Recognition

463

~ AND f is ~ …. AND f is ~ then emotion class is c. Rc: if f1 is A A2 Am 2 m 1 Here, fi for i=1 to m are m-measurements (feature value) in the general type-2 ~ ~ ~ respectively, given by fuzzy sets A1 , A2 , …, A m ~ Ai = [ μ A~ ( f i ), μ A~ ( f i )], ∀i .

(2)

Since an emotion is characterized by all of these m features, to find the overall support of the m features (m measurements made for the unknown subject) to the emotion class c represented by the n primary memberships, we use the fuzzy meet operation S cmin = Min{

mod

μ A~ ( f1/ ), mod μ A~ ( f 2/ ),.....,mod μ A~ ( f m/ ), }

mod

S cmax = Min{

μ A~ ( f1/ ),

mod

μ A~ ( f 2/ ),.....,

mod

μ A~ ( f m/ )}

(3) (4)

Thus we can say that the unknown subject is experiencing the emotion class c at least min max to the extent sc , and at most to the extent sc . To reduce the non-specificity associated with the interval Sc-i, the most conservative approach would be to use lower bound, while the most liberal view would to use the upper bound of the interval as the support for the class c. In absence of any additional information, a balanced approach would be to use center of the interval as the support for the class c by the n primary memberships to the unknown subject [8]. We compute the centre, Sc of the interval Sc-i.Thus Sc is the degree of support that the unknown facial expression is in emotion class c. Sc= (scmin + scmax)/2.

(5)

Now to predict the emotion of a person from his facial expression, we determine Sc for each emotion class. Presuming that there exist k emotion classes, let us denote them by S1, S2,…., Sk for emotion class 1, 2,…, k, respectively. Since a given facial expression may convey different emotions with different degrees, we resolve the conflict by ranking the Si for i = 1 to k, and thus determine the emotion class r, for which Sr >= Si for all i following the Rule Rc. To make the algorithm robust, we consider association of fuzzy encoded measurements with emotion class by considering the weakest reliability of the joint occurrence of the fuzzy measurements, and identify the winning emotion class having this measure of reliability superseding the same of other emotion classes.

3

Methodology

We now briefly discuss the main steps involved in fuzzy face space construction based on the measurements of m facial features for n-subjects, each having l instances of facial expression for a particular emotion. We need to classify a facial expression of an unknown person into one of k emotion classes.

464

1.

2.

3.

A. Halder et al.

We extract m facial features for n subjects, each having l instances of facial expression for a particular emotion. The above features are extracted for kemotion classes. We construct a fuzzy face space for each emotion class separately. The fuzzy face space for an emotion class comprises a set of n primary membership and secondary membership distributions for each feature. Thus we have m groups of n-primary as well as secondary membership distributions. Each membership curve is constructed from l-facial instances of a subject attempted to exhibit a particular emotion in her facial expression by acting. For a given feature fi/, we consult each primary and secondary membership curve under a given emotion class, and take the product of primary and secondary membership at fi= fi/. The resulting membership value obtained for the membership curves for the subject w is given by mod

μ A~w ( f i / ) = μ A~w ( f i / ) × μ ( f i / , μ A~w ( f i / ))

Now, for w = 1 to n, we evaluate the maximum values of [

4.

mod

μ A~ ( f i ), /

mod

mod

mod

(6)

μ Aw~ ( f i / ) , and thus obtain the minimum and

μ Aw~ ( f i / ) , to

obtain a range of uncertainty

μ A~ ( f i )] . This is repeated for all features under each emotion /

class. Now for an emotion class j, we take fuzzy meet operation over the ranges for each feature to evaluate the range of uncertainty for individual emotion class. The meet operation here is computed by taking cumulative t-norm of mod min mod μ A~ ( f i / ) separately for i= 1 to m, and thus obtaining Sj and μ ~ ( f i / ) and A

5. 6.

4

Sjmax respectively. The support of the j-th emotion class to the measurements is evaluated by taking average of Sjmin and Sjmax, and defining the result by Sj. Now by using classifier rule, we determine the maximum support offered by all the k emotion classes, and declare the unknown facial expression to have emotion r, if Sr > Si for all emotion class i= 1 to k.

Experimental Details

In this section, we present the experimental details of emotion recognition using the principles introduced in section 2 and 3. We here consider 5 emotion classes, (i.e., k=5) including anger, fear, disgust, happiness and relaxation. The experiment is conducted with two sets of subjects: a) the first set of n (=10) subjects is considered for designing the fuzzy face-space and, b) the other set of 30 facial expressions taken from 6 unknown subjects are considered to validate the result of the proposed emotion classification scheme. Five facial features, (i.e., m=5) have been used here to design the type-2 fuzzy face-space. We now briefly outline the main steps.

Application of General Type-2 Fuzzy Set in Emotion Recognition

4.1

465

Feature Extraction

Feature extraction is a fundamental step in emotion recognition. Existing research results [9],[10] reveal that the most important facial regions responsible for the manifestation of emotion are the eyes and the lips. This motivated us to select the following features: Left Eye Opening (EOL), Right Eye Opening (EOR), Distance between the Lower Eyelid to the Eyebrow for the Left Eye (LEEL), Distance between the Lower Eyelid to Eyebrow for the Right Eye (LEER), and the Maximum Mouth opening (MO) including the lower and the upper lips. Fig. 1 explains the above facial features on a selected facial image.

LEER

LEEL

EOR

EOL

MO

Fig. 1. Facial Features

4.2

Creating the Type-2 Fuzzy Face-Space

The Type-2 fuzzy face-space contains the primary and corresponding secondary membership distributions for each facial feature. Since we have 5 facial features and the experiment includes 5 distinct emotions of 10 subjects, we obtain 10×5×5=250 primary as well as secondary membership curves. To compute primary memberships, 10 instances of a given emotion is used. These 250 membership curves are grouped into 25 heads, each containing 10 membership curves of ten subjects for a specific feature for a given emotion. Fig. 2. (a) gives an illustration of one such group of 10 membership distributions for the feature EOL for the emotion: disgust. 1 0.9

1 primary memberships

Primary memberships -->

0.8 0.7 0.6 0.5 0.4 0.3

0.8 0.6 0.4 0.2 0 1

0.2

1.5 0.5

0.1 0 0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

secondary memberships

1 feature 0

0.5

Feature -->

Fig. 2. (a). Membership distributions for emotion disgust and feature EOL (b) Secondary Membership curve of Subject1

For each primary membership distribution, we have a corresponding secondary membership distribution. Thus we obtain 250 secondary membership distributions. One illustrative type-2 secondary distributions of subject 1 for the feature EOL for the

466

A. Halder et al.

emotion disgust are given in Fig. 2. (b) The axes in the figure represent feature (EOL), primary and secondary membership values as indicated. 4.3

Emotion Recognition of an Unknown Facial Expression

The emotion recognition problem addressed here attempts to determine the emotion of an unknown person from her facial expression. To keep the measurements in an emotional expression normalized and free from distance variation from the camera focal plane, we construct a bounding box, covering only the face region, and the reciprocal of the diagonal of the bounding box is used as a scale factor for normalization of the measurements. The normalized features obtained from Fig.3 are enlisted in Table 1. We now briefly explain the experimental results obtained by our proposed method.

 Fig. 3. Facial Image of an unknown subject Table 1. Extracted Features of Fig. 3 EOL 0.026

EOR 0.026

MO 0.135

LEEL 0.115

LEER 0.115

The GT2FS based recognition scheme considers a fuzzy face space of 5 sets of 10 primary membership distributions as in Fig. 2 (a), and the corresponding secondary membership distributions to the individual primary membership distribution of 5 features obtained from facial expressions carrying 5 distinct emotions for 10 different subjects are determined using curves like Fig. 2 (b). Table 2. Calculated Type-2 Membership Values For Feature :EOL, Emotion: Disgust Feature

EOL

Primary Memberships (μpri) 0.65 0.10 0.15 0.45 0.18 0.55 0.08 0.41 0.16 0.12

Secondary memberships (μsec) 0.72 0.55 0.58 0.68 0.56 0.68 0.55 0.63 0.53 0.59

μmod=μpri × μsec

0.468 0.055 0.087 0.306 0.1008 0.374 0.044 0.2583 0.0848 0.0708

Range (min{ μmod},max{ μmod })

0.044-0.468

Application of General Type-2 Fuzzy Set in Emotion Recognition

467

Table 3. Calculated Feature Ranges and Centre Value For Each Emotion

Range of Secondary Membership for Features Emotion EOL

EOR

MO

LEEL

LEER

0.0034-0.814 0.0029-0.781 0-0.39 0.-0.283 0.054-0.473 0.057-0.511

Range Scj after fuzzy Meet operation (centre) 0-0.14(0.07) 0-0(0) 0-0.275(0.1375)

Anger Disgust Fear

0-0.14 0 - 0.17 0.39-0.964 0.044-0.468 0.041-0.531 0-0 0 - 0.298 0-0.275 0.04-0.742

Happy

0 - 0.555

0-0.604

0.573-0.910 0.133-0.851

0.3214-0.7213 0-0.555(0.2775)

Relaxed

0 - 0.132

0-0.221

0-0

0.046-0.552

0.013-0.458

0-0(0)

Table 2 provides the summary of the primary and secondary memberships obtained for EOL for the emotion: disgust. For each feature we obtain 5 Tables like Table 2, each one for a given emotion. Thus for 5 features, we would have altogether 25 such tables. In Table 2, we also computed the product of primary and secondary memberships, and then obtain the minimum and maximum of the product to determine its range, as indicated in the last column of Table 2. The range for each feature corresponding to individual emotions is given in Table 3. For example, the entry (0-0.14) corresponding to the row Anger and column EOL, gives an idea about the extent of the EOL for the unknown subject matches with known subjects from the emotion class Anger. The results of computing fuzzy meet operation over the range of individual features taken from facial expressions of the subjects under same emotional condition are given in Table 3. The average of the ranges along with its centre value is also given in Table 3. It is observed that the centre has the largest value (=0.2775) for the emotion: happiness

5

Conclusion

The paper employs GT2FS-based automatic emotion recognition of an unknown facial expression, when the background knowledge about a large face database with known emotion class are available. The GT2FS-based recognition scheme requires type-2 secondary membership distributions, a computation of which by an evolutionary approach is also provided. The scheme first construct a fuzzy face space, and then infer the emotion class of the unknown facial expression by determining the maximum support of the individual emotion classes using the pre-constructed fuzzy face space. The class with the highest support is regarded as the emotion of the unknown facial expression. The scheme, however, takes care of both the inter- and intrasubject level uncertainty, and thus offers a higher classification accuracy for the same set of features. Experimental analysis confirms that the classification accuracy of emotion by employing GT2FS is 96.67%.

468

A. Halder et al.

References 1. Uwechue, O.A., Pandya, S.A.: Human Face Recognition Using Third-Order Synthetic Neural Networks. Kluwer, Boston (1997) 2. Biswas, B., Mukherjee, A.K., Konar, A.: Matching of digital images using fuzzy logic. AMSE Publication 35(2), 7–11 (1995) 3. Bhavsar, A., Patel, H.M.: Facial Expression Recognition Using Neural Classifier and Fuzzy Mapping. In: IEEE Indicon 2005 Conference, Chennai, India (2005) 4. Guo, Y., Gao, H.: Emotion Recognition System in Images Based on Fuzzy Neural Network and HMM. In: Proc. 5th IEEE Int. Conf. on Cognitive Informatics (ICCI 2006). IEEE (2006) 5. Rizon, M., Karthigayan, M., Yaacob, S., Nagarajan, R.: Japanese face emotions classification using lip features. In: Geometric Modelling and Imaging (GMAI 2007), Universiti Malaysia Perlis, Jejawi, Perlis, Malaysia. IEEE (2007) 6. Kobayashi, H., Hara, F.: Measurement of the strength of six basic facial expressions by neural network. Trans. Jpn. Soc. Mech. Eng. (C) 59(567), 177–183 (1993) 7. Ekman, P., Friesen, W.V.: Unmasking the Face: A Guide to Recognizing Emotions From Facial Clues. Prentice-Hall, Englewood Cliffs (1975) 8. Mendel, J.M.: On the importance of interval sets in type-2 fuzzy logic systems. In: Proc. Joint 9th IFSA World Congress 20th NAFIPS Int. Conf., Vancouver, BC, Canada, July 2528, pp. 1647–1652 (2001) 9. Chakraborty, A., Konar, A., Chakraborty, U.K., Chatterjee, A.: Emotion Recognition From Facial Expressions and Its Control Using Fuzzy Logic. IEEE Transactions on Systems, Man and Cybernetics (2009) 10. Das, S., Halder, A., Bhowmik, P., Chakraborty, A., Konar, A., Nagar, A.K.: Voice and Facial Expression Based Classification of Emotion Using Linear Support Vector. In: 2009 Second International Conference on Developments in eSystems Engineering. IEEE (2009)

Design of a Control System for Hydraulic Cylinders of a Sluice Gate Using a Fuzzy Sliding Algorithm Wu-Yin Hui and Byung-Jae Choi School of Electronic Engineering, Daegu University, Jillyang, Gyeongsan, Gyeongbuk 712-714, Korea [email protected], [email protected]

Abstract. The hydraulic system has the characteristic of high pressure, high speed, high efficiency, low noise, and high durability. Hydraulic has a wide range of applications. In this paper, we consider two control objectives of this hydraulic system. The first is to push or pull the sluice gate with an exact speed to an exact position. The second is to remove the displacement different between two cylinders when they are moving together. In order to design the controller for this system, we propose a fuzzy-sliding position control system. And to remove the difference between two cylinders, we propose the fuzzy PI synchronous controller. We show some simulation results of its availability.

Keywords: Hydraulic Cylinders, Fuzzy Logic Control, Position Controller, Synchronous Controller, Sluice Gate.

1

Introduction

The hydraulic system is one of the important mechanical systems. The hydraulic system has the characteristic of high pressure, high speed, high efficiency, low noise, and high durability. Hydraulic has a wide range of applications. For instance, the bulldozer and excavator in construction machinery, hydraulic jack in building machinery, combine harvester and tractor in agricultural machinery, and the ABS system in automobile are all examples of hydraulic system applications. The hydraulic cylinder system has been discussed in many papers. It can convert hydraulic energy into mechanical, and it moves in straight reciprocating motion. In order to control the position of a cylinder, a servo valve can be used. Servo valve can change the supply flow by changing the area of orifice. A servo-hydraulic system can be used to provide large processing force and has a good positioning capability, but this system is complex and highly nonlinear. B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 469–476, 2011. © Springer-Verlag Berlin Heidelberg 2011

470

W.-Y. Hui and B.-J. Choi

In this study, the synchronous position control of a sluice gate with two cylinders is considered. In order to push the cylinders to an exact position with an exact speed and to remove the difference between the displacements of the two cylinders, two sliding-fuzzy controllers and two fuzzy PI controllers have been designed. We performed simulated to show the effectiveness of the proposed control system design. The result that combined and emulated the control system in Matlab –Simulink and AMESim (Advanced Modeling for Simulation of Engineering) coincided with the target.

2

System Description

In this system, which is shown Figure 1, the input of the hydraulic servo system is current. The spool of hydraulic servo valve is based on the input current. The spool movement opens the control orifice. After opening the orifice, the hydraulic can flow through the pass orifice and pass in or out of the cylinder chambers. The supply flow and the supply pressure are based on the open area of orifice depending on the current. The hydraulic oil that flows through orifice creates pressure difference between head side and load side of the piston. And this pressure difference pushes or pulls the sluicegate. Power is produced by the linear motion of the piston in cylinder.

Fig. 1. Composition of hydraulic servo system

Design of a Control System for Hydraulic Cylinders

471

Fig. 2. The overall structure of a hydraulic servo system

3

Design of Fuzzy Sliding Controller

We consider two control objectives: the first objective is to push or pull the sluice gate with an exact speed to an exact position. The second objective is to remove the displacement difference between two cylinders when they are moving together. In order to design the controller for this system, we designed a fuzzy-sliding position control system. And a fuzzy PI synchronous controller is designed to remove the difference between two cylinders. Although there are two controllers, there is only one output that is calculated by the output values of both the position controller and synchronous controller. 3.1

Fuzzy Sliding Position Controller

In the sliding model, the control law contains both the equivalent control law and the switching law. The equivalent control law keeps the system state on the sliding surface, and the switching law can constrain the system state to move on the sliding surface. By using a fuzzy algorithm, the system contains both the equivalent control law and the switching law. And the fuzzy membership functions are shown in Fig. 3.

Fig. 3. The input and output membership functions for the fuzzy logic controller

472

3.2

W.-Y. Hui and B.-J. Choi

Fuzzy PI Synchronous Controller

Another controller is used to remove the displacement difference between the two cylinders. However, some problems can take place when we use two controllers together. In this paper, the synchronous controller is designed with three inputs, the first and the second ones are the difference between two cylinders and its differential coefficient, while the last one is the gap between the reference signal and displacement signal of each cylinder. In this case, when the displacement signal is close to the reference signal, the output of synchronous controller will be reduced. Moreover, even when the cylinders displacement is larger or equal to the reference signal, the output of the fuzzy PI synchronous controller is ZO. The membership functions of the synchronous controller are in following figure and the rules are shown in following table.

(a) Input “e”

(c) Input “de/dt”

(b) Input “d”

(d) Output

Fig. 4. Membership functions of the input for the synchronous controller

Design of a Control System for Hydraulic Cylinders

473

Table 1. Rule table for synchronous controller

3.3

Control System Output Algorithm

In this paper, two controllers have been designed for the control system. The output is calculated using output values of both position controller and synchronous controller. The control system is shown in following figure.

474

W.-Y. Hui and B.-J. Choi

Fig. 5. The system structure of PID control system

4

Simulation Results

The system has been combined and emulated in both Matlab –Simulink and AMSim. The controller was modeled by Matlab –Simulink, and the control plant was constructed in AMSim. The overall strure is shown in following Figure.

Fig. 6. The model of the overall control system and controlled process

4.1

Simulation for Extending the Stroke of 120 Seconds

The speed of cylinders is set to 4.7mm/s. the simulation results of the output of controllers are shown in following figure.

Design of a Control System for Hydraulic Cylinders

475

Fig. 7. Displacement for cylinder A and B, and its difference (at the extending stroke)

4.2

Simulation for Retracting the Stroke of 120 Seconds

The speed of cylinders is set to -2.3mm/s. the simulation results of the output of controllers are shown in following figure.

Fig. 8. Displacement for cylinder A and B, and its difference (at the retracting stroke)

5

Conclusion

In this paper, the fuzzy sliding position controller and fuzzy PI synchronous controller have been considered. The synchronous controller could regulate the output based on the difference between the reference signal and the position signal. Form the simulation result, the cylinders could track with the reference signal correctly. The difference of cylinders was bound well.

References 1. Zhao, Y.-X., Chen, X.-D., Chen, X.: Repeatability Analysis of a Vulcanizer Hydraulider System using Fuzzy Arithmetic. In: International Conference on Computational Intelligence for Measurement System and Application

476

W.-Y. Hui and B.-J. Choi

2. Kim, J.H.: A Suggestion of Nonlinear Fuzzy PID Controller to Improve Transient Responses of Nonlinear or Uncertain System. International Journal of Fuzzy Logic and Intelligent System 5(4), 87–100 (1995) 3. Lee, C.C.: Fuzzy Logic in Control System. IEEE Trans. On System, Man and Cybernetica 20(2) (1990) 4. Hung, H.T., Kwan, A.K.: A Study on the Position Control of Hydraulin Cylinder Driven by Hydraulic Transformer using Disturbance Observer. In: International Conference on Control, Automation and Systems (2008)

Rough Sets for Selection of Functionally Diverse Genes from Microarray Data Sushmita Paul and Pradipta Maji Machine Intelligence Unit, Indian Statistical Institute, Kolkata, India {sushmita t,pmaji}@isical.ac.in

Abstract. Selection of reliable genes from a huge gene expression data containing high intergene correlation is essential to carry out a diagnostic test and successful treatment. In this regard, a rough set based gene selection algorithm is reported, which selects a set of genes by maximizing the relevance and significance of the selected genes. A gene ontology-based similarity measure is proposed to analyze the functional diversity of the selected genes. It also helps to analyze the effectiveness of different gene selection methods. The performance of the rough set based gene selection algorithm, along with a comparison with other gene selection methods, is studied using the predictive accuracy of K-nearest neighbor rule and support vector machine on two cancer and one arthritis microarray data sets. An important finding is that the rough set based gene selection algorithm selects more functionally diverse set of genes than the existing algorithms. Keywords: Rough sets, Gene selection, Functional diversity, Gene ontology.

1 Introduction A microarray gene expression data set can be represented by an expression table, where each row corresponds to one particular gene, each column to a sample or time point, and each entry of the matrix is the measured expression level of a particular gene in a sample or time point, respectively [2]. However, among the large amount of genes, only a small fraction is desirable in developing gene expression based diagnostic tools for delivering precise, reliable, and interpretable results. Hence, identifying a reduced set of most relevant and significant genes is the goal of gene selection. The small number of training samples and a large number of genes make it a challenging problem for discriminant analysis of microarray data. In this background, Loennstedt and Speed [6] developed a gene selection algorithm based on the empirical bayes moderated t-statistic that ranks genes by testing whether all pairwise contrasts between different classes are zero. On the other hand, significance analysis of microarrays (SAM) method [9] assigns a score to each gene on the basis of change in gene expression relative to the standard deviation of repeated measurements. All of these methods are univariate and donot consider the effect of one gene on another gene. Hence, the results obtained by these methods contain redundant gene set and may not be able to generate functionally diverse gene set. The main objective of the current research work is to select a set of genes, which are functionally diverse and contain least redundant information. In this regard, a rough set B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 477–484, 2011. c Springer-Verlag Berlin Heidelberg 2011 

478

S. Paul and P. Maji

based gene selection algorithm [7] is reported to select a set of genes from microarray gene expression data by maximizing both relevance and significance of the selected genes. It employs rough set theory to compute the relevance and significance of the genes. A method is introduced to identify the optimum values of different parameters used in the rough set based gene selection algorithm. The performance of the rough set based gene selection algorithm is compared with that of existing approaches using the predictive accuracy of K-nearest neighbor rule and support vector machine on three microarray data sets. A gene ontology-based similarity assessment has been done for the genes selected by different gene selection algorithms to understand the functional diversity of genes.

2 Rough Set Based Gene Selection Algorithm In real data analysis such as microarray data, the data set may contain a number of insignificant genes. The presence of such irrelevant and insignificant genes may lead to a reduction in the useful information. Ideally, the selected genes should have high relevance with the classes and high significance in the gene set and thus the set enhances the predictive capability. Accordingly, a measure is required that can enhance the effectiveness of gene set. In this regard, a gene selection algorithm is developed in [7], which is based on the theory of rough sets to select the relevant and significant genes from high dimensional microarray data. Let C = {A1 , · · · , Ai , · · · , A j , · · · , Am } denotes the set of m features or genes of a given microarray data set and S is the set of selected genes. Define fˆ(Ai , D) as the relevance of the gene Ai with respect to the class labels D while f˜(Ai , A j ) as the significance of the gene A j with respect to the gene Ai . Hence, the total relevance of all selected genes and the total significance among the selected genes are given by Jrelev =

∑

Ai ∈S

fˆ(Ai , D); Jsignf =

∑

Ai =A j ∈S

f˜(Ai , A j ).

Therefore, the problem of selecting a set S of relevant and significant genes from the whole set C of m genes is equivalent to maximize both Jrelev and Jsignf , that is, to maximize the objective function J , where J = Jrelev + β Jsignf =

∑

Ai ∈S

fˆ(Ai , D) + β

∑

f˜(Ai , A j )

(1)

Ai =A j ∈S

where β is a weight parameter. Both the relevance and significance are calculated based on the theory of rough set. The main steps of this algorithm are presented in [7]. 2.1 Generation of Equivalence Classes To calculate relevance and significance of genes using rough set theory, the continuous expression values of a gene have to be divided into several discrete partitions to generate equivalence classes. However, the inherent error that exists in conventional discretization processes is of major concern in the computation of relevance and significance

Rough Sets for Selection of Functionally Diverse Genes

479

of continuous valued genes. To address this problem, a fuzzy set based discretization method is used in [7] to generate equivalence classes required to compute both relevance and significance of genes using rough set theory. Given a finite set U, C is a fuzzy condition attribute set in U, which generates a fuzzy equivalence partition on U. If c denotes the number of fuzzy equivalence classes generated by the fuzzy equivalence relation and n is the number of objects in U, then a matrix MC of size (c × n) containing elements {μiCj } can be generated MC = [μiCj ], which is denoted by ⎛

C μ11 ⎜ μC 21 MC = ⎜ ⎝··· C μc1

C μ12 C μ22 ··· C μc2

⎞ C · · · μ1n C ⎟ · · · μ2n ⎟ ··· ··· ⎠ C · · · μcn

(2)

where μiCj ∈ [0, 1] represents the membership of object x j in the ith fuzzy equivalence partition or class Fi . The above axioms should hold for every fuzzy equivalence partition, which correspond to the requirement that an equivalence class is nonempty. In the rough set based gene selection method, the π function in one dimensional form is used to assign membership values to different fuzzy equivalence classes for the input genes. A fuzzy set with membership function π (x; c, ¯ σ ) represents a set of points clustered around c, ¯ where ⎧ ||x−c|| ¯ ⎪ ¯ ≤σ ⎨ 2(1 − σ )2 for σ2 ≤ ||x − c|| ¯ 2 π (x; c, ¯ σ ) = 1 − 2( ||x−c|| (3) ¯ ≤ σ2 σ ) for 0 ≤ ||x − c|| ⎪ ⎩ 0 otherwise where σ > 0 is the radius of the π function with c¯ as the central point and || · || denotes the Euclidean norm. The (c × n) matrix MAi , corresponding to the ith gene Ai , can be calculated from the c-fuzzy equivalence classes of the objects x = {x1 , · · · , x j , · · · , xn }, where π (x j ; c¯k , σk ) μkAji = c . (4) π (x ; c ¯ , σ ) ∑ j l l l=1

In effect, each position μkAji of the matrix MAi must satisfy the following conditions:

μkAj i ∈ [0, 1]; s = arg

c

∑ μkAji = 1, ∀ j and for any value of k, if

k=1 max{ μkAji }, j

then max{ μkAji } = max{ μlsAi } > 0. j

l

After the generation of the matrix MAi corresponding to the gene Ai , the object x j is assigned to one of the c equivalence classes based on the maximum value of memberships of the object in different equivalence classes that follows next: x j ∈ Fp ,

where p = arg max{μkAji }. k

480

S. Paul and P. Maji

Each input real valued gene in quantitative form can be assigned to different fuzzy equivalence classes in terms of membership values using the π fuzzy set with appropriate c¯ and σ . In the rough set based gene selection algorithm, three fuzzy equivalence classes (c=3), namely, low, medium, and high are considered. These three equivalence classes correspond to under-expression, base-line, and over-expression of continuous valued genes, respectively and also hold the following relations: c¯1 = c¯low (Ai ); c¯2 = c¯medium (Ai ); c¯3 = c¯high (Ai ); σ1 = σlow (Ai ); σ2 = σmedium (Ai ); σ3 = σhigh (Ai ). The parameters c¯ and σ of each π fuzzy set are computed according to the following procedure [8]. Let m¯ i be the mean of the objects x = {x1 , · · · , x j , · · · , xn } along the ith gene Ai . Then m¯ il and m¯ ih are defined as the mean along the ith gene of the objects having co-ordinate values in the range [Aimin , m¯ i ) and (m¯ i , Aimax ], respectively, where Aimax and Aimin denote the upper and lower bounds of the dynamic range of gene Ai for the training set. For three fuzzy sets low, medium, and high, the centers and corresponding radii are computed as follows: A ; B σlow (Ai ) = 2[c¯medium (Ai ) − c¯low (Ai )]; σhigh (Ai ) = 2[c¯high (Ai ) − c¯medium(Ai )]; c¯low (Ai ) = m¯ il ; c¯medium (Ai ) = m¯ i ; c¯high (Ai ) = m¯ ih ; σmedium (Ai ) = η ×

where A = {σlow (Ai )(Aimax − cmedium (Ai )) + σhigh (Ai )(cmedium (Ai ) − Aimin )}; B = {Aimax − Aimin } where η is a multiplicative parameter controlling the extent of the overlapping between three fuzzy equivalence classes low and medium or medium and high.

3 Experimental Results and Discussions The performance of the rough set based gene selection method (MRMS) [7] is extensively studied and compared with that of some existing algorithms, namely, Hall’s combinatorial feature selection algorithm (CFS) [5], Ebayes, Ensemble, partial least squares cross-validation method (PLSCV) [5], random forest feature selection (RFMDA) [1], and SAM [9]. To analyze the performance of different gene selection methods, the experimentation is done on two cancer (Breast, and Leukemia) and one arthritis (RAOA) microarray data sets. For each data set, fifty top-ranked genes are selected for analysis. The performance of different gene selection methods is evaluated by using the classification accuracy of K-nearest neighbor (K-NN) rule and support vector machine (SVM). To compute the prediction accuracy of both SVM and K-NN rule, the leave-one-out cross-validation (LOOCV) is performed on each microarray data set. The biological evaluation of these methods is also conducted using G-Sesame [3], which is developed using the concept of gene ontology.

Rough Sets for Selection of Functionally Diverse Genes

481

3.1 Optimum Values of Parameters The weight parameter β in (1) plays an important role to reduce redundancy in between genes present in microarray data. In effect, it has a direct influence on the performance of the MRMS algorithm. Also, the MRMS algorithm depends on multiplicative parameter η . Let S={β , η } be the set of parameters and S∗ ={β ∗ , η ∗ } is the set of optimal parameters. To find out the optimum set S∗ , containing optimum values of β ∗ and η ∗ , the support vector machine [10] and K-nearest neighbor (K-NN) [4] are used. Support Vector Machine. The support vector machine (SVM) [10] is a margin classifier that draws an optimal hyperplane in the feature vector space; this defines a boundary that maximizes the margin between data samples in different classes, therefore leading to good generalization properties. A key factor in the SVM is to use kernels to construct nonlinear decision boundary. In the present work, linear kernels are used. K-Nearest Neighbor Rule. The K-nearest neighbor (K-NN) rule [4] is used for evaluating the effectiveness of the reduced gene set for classification. It classifies samples based on closest training samples in the feature space. A sample is classified by a majority vote of its K-neighbors, with the sample being assigned to the class most common amongst its K-nearest neighbors. The value of K, chosen for the K-NN, is the square root of the number of samples in training set. For three microarray data sets, the value of β is varied from 0.0 to 1.0, while the value of η is varied from 0.5 to 2.0. The optimum values of β ∗ and η ∗ for each microarray data set are obtained using the following relation: S∗ = arg max{Classification accuracy of D}; where D is either SVM or KNN. (5) S

The optimum sets S∗ containing optimum values of β and η obtained using (5) are {0.3, 0.8}, {0.1, 1.5}, and {0.6, 0.8} for Breast, Leukemia, and RAOA data sets, respectively using the SVM, while {0.6, 1.2}, {0.1, 1.7}, and {0.7, 0.9} for Breast, Leukemia, and RAOA data sets, respectively using the KNN. 3.2 Comparative Performance Analysis of Different Algorithms The best results of different algorithms on three microarray data sets are presented in Table 1. The classification accuracy of both SVM and K-NN of the MRMS is better compared to other gene selection algorithms in all cases. The best performance of the MRMS algorithm, in terms of accuracy of both SVM and K-NN, is achieved due to the maximization of relevance and significance of genes simultaneously. 3.3 Gene Ontology Based Analysis to Study Independency of Selected Genes The web based tool G-SESAME [3] is used to measure functional similarity between a pair of genes obtained by various gene selection algorithms. This tool quantifies functional similarity between a pair of genes based on gene annotation information from heterogeneous data sources. The value of similarity ranges from zero to one, nearer

482

S. Paul and P. Maji Table 1. Performance of Different Gene Selection Algorithms Data Sets

Methods/ Algorithms CFS EBayes ENSEMBLE Breast PLSCV RFMDA SAM MRMS CFS EBayes ENSEMBLE Leukemia PLSCV RFMDA SAM MRMS CFS EBayes ENSEMBLE RAOA PLSCV RFMDA SAM MRMS

SVM Accuracy Number of genes 97.96 28 93.88 4 95.92 6 95.92 6 95.92 5 93.88 7 100.00 6 100.00 7 97.22 36 98.61 15 100.00 47 97.22 7 98.61 43 100.00 4 100.00 7 93.33 1 100.00 7 100.00 7 100.00 7 100.00 7 100.00 4

K-NN Accuracy Number of genes 97.96 20 93.88 24 91.84 8 95.92 9 93.88 15 93.88 18 100.00 6 98.61 21 97.22 34 97.22 14 97.22 38 97.22 11 97.22 45 100.00 3 93.33 2 90.00 1 96.67 48 96.67 48 96.67 48 96.67 48 100.00 8

the value to one higher the functional similarity between the pair of genes. Given two genes G1 and G2 and their annotated GO terms GO1 ={go11,go12 ,...,go1m} and GO2 = {go21,go22,...,go2n }, their functional similarity is define as: m

n

∑ Sim(go1i, GO2 ) + ∑ Sim(go2 j , GO1 )

Sim(G1, G2) =

1=1

1= j

m+n

(6)

In the present work, the G-SESAME is used to quantify functional diversity present in between selected genes. The numbers of gene pairs obtained by G-SESAME tool for a dataset in all methods and in all ontologies are not uniform. To make the comparison between all methods, a measure called degree of functional diversity (DoFD), is proposed next. For a given distribution of similarity (C), the DoFD for each method can be calculated by 5

DoFD(C) = ∑ wi ni

(7)

i=1

where wi ={0.50,0.25,0.125,0.0625,0.0625} is weight parameter and ni is the fraction of each category of similarity distribution table, which is obtained by taking ratio of frequency and gene pairs number. The frequency is calculated by counting the number of gene pairs which occur in a particular category. Here, five categories are generated. Gene pairs having semantic similarity in range of 0.0 to 0.5, signifies that the gene pairs are functionally very diverse. Higher the value of DoFD means higher functional diversity. This tool is applied on gene sets selected by CFS, Ebayes, Ensemble, PLSCV, RFMDA, SAM, and MRMS methods on three datasets. The functional diversity is calculated for three ontologies, namely, biological processes (BPs), molecular functions

Rough Sets for Selection of Functionally Diverse Genes

483

Table 2. Degree of Functional Diversity and Distribution of Best Selected Genes Microarray Different Data Sets Ontologies

MF

Breast

BP

CC

MF

Leukemia

BP

CC

Methods/ Algorithms CFS(483) Ebayes(1015) Ensemble(924) PLSCV(648) RFMDA(1003) SAM(885) MRMS(SVM)(360) MRMS(K-NN)(220) CFS(525) Ebayes(1060) Ensemble(935) PLSCV(682) RFMDA(1053) SAM(969) MRMS(SVM)(395) MRMS(K-NN)(220) CFS(518) Ebayes(1036) Ensemble(924) PLSCV(648) RFMDA(961) SAM(924) MRMS(SVM)(360) MRMS(K-NN)(220) CFS(435) Ebayes(1036) Ensemble(891) PLSCV(806) RFMDA(1036) SAM(999) MRMS(SVM)(703) MRMS(K-NN)(616) CFS(432) Ebayes(990) Ensemble(868) PLSCV(806) RFMDA(1003) SAM(957) MRMS(SVM)(675) MRMS(K-NN)(646) CFS(434) Ebayes(1036) Ensemble(918) PLSCV(783) RFMDA(1045) SAM(1007) MRMS(SVM)(585) MRMS(K-NN)(585)

DoFD 0.672 0.709 0.641 0.662 0.635 0.755 0.964 0.980 0.833 0.865 0.864 0.815 0.836 0.874 0.963 0.980 0.551 0.596 0.642 0.684 0.670 0.632 0.940 0.973 0.447 0.686 0.663 0.550 0.621 0.646 0.840 0.881 0.869 0.894 0.897 0.863 0.907 0.894 0.935 0.950 0.501 0.653 0.614 0.666 0.613 0.611 0.864 0.878

0.0-0.2 Freq. Frac. 254 0.526 596 0.587 460 0.498 359 0.554 502 0.500 580 0.655 337 0.940 214 0.970 365 0.695 798 0.753 708 0.757 464 0.680 739 0.702 749 0.773 369 0.930 214 0.970 180 0.347 428 0.413 439 0.475 356 0.549 522 0.543 441 0.477 326 0.910 212 0.960 100 0.230 603 0.582 497 0.558 341 0.423 496 0.479 540 0.541 549 0.780 510 0.830 325 0.752 799 0.807 703 0.810 597 0.741 832 0.830 772 0.807 603 0.890 589 0.910 96 0.221 510 0.492 389 0.424 408 0.521 443 0.424 426 0.423 480 0.820 480 0.820

Similarity Distribution 0.2-0.4 0.4-0.6 0.6-0.8 Freq. Frac. Freq. Frac. Freq. Frac. 69 0.143 130 0.269 21 0.043 117 0.115 220 0.217 42 0.041 121 0.131 229 0.248 69 0.075 48 0.074 128 0.198 75 0.116 111 0.111 247 0.246 99 0.099 78 0.088 164 0.185 24 0.027 15 0.040 4 0.010 4 0.010 5 0.020 1 0.000 0 0.000 131 0.250 27 0.051 2 0.004 218 0.206 36 0.034 6 0.006 178 0.190 40 0.043 6 0.006 167 0.245 16 0.023 11 0.016 258 0.245 41 0.039 12 0.011 177 0.183 31 0.032 8 0.008 23 0.060 2 0.010 1 0.000 5 0.020 1 0.000 0 0.000 123 0.237 135 0.261 64 0.124 224 0.216 234 0.226 122 0.118 202 0.219 139 0.150 104 0.113 115 0.177 59 0.091 60 0.093 113 0.118 194 0.202 106 0.110 165 0.179 167 0.181 120 0.130 14 0.040 12 0.030 5 0.010 5 0.020 3 0.010 0 0.000 91 0.209 148 0.340 73 0.168 88 0.085 163 0.157 140 0.135 81 0.091 116 0.130 150 0.168 53 0.066 195 0.242 103 0.128 140 0.135 216 0.208 148 0.143 71 0.071 172 0.172 164 0.164 50 0.070 38 0.050 53 0.080 34 0.060 37 0.060 27 0.040 95 0.220 10 0.023 1 0.002 158 0.160 26 0.026 6 0.006 140 0.161 17 0.020 7 0.008 189 0.234 14 0.017 5 0.006 141 0.141 24 0.024 5 0.005 155 0.162 22 0.023 7 0.007 55 0.080 13 0.020 3 0.000 43 0.070 10 0.020 3 0.000 174 0.401 112 0.258 47 0.108 204 0.197 191 0.184 106 0.102 221 0.241 203 0.221 87 0.095 135 0.172 127 0.162 100 0.128 251 0.240 227 0.217 104 0.100 245 0.243 198 0.197 110 0.109 25 0.040 38 0.060 25 0.040 45 0.080 23 0.040 24 0.040

0.8-1.0 Freq. Frac. 9 0.019 40 0.039 45 0.049 38 0.059 44 0.044 39 0.044 0 0.000 0 0.000 0 0.000 2 0.002 3 0.003 24 0.035 3 0.003 4 0.004 0 0.000 0 0.000 16 0.031 28 0.027 40 0.043 58 0.090 26 0.027 31 0.034 3 0.010 0 0.000 23 0.053 42 0.041 47 0.053 114 0.141 36 0.035 52 0.052 13 0.020 8 0.010 1 0.002 1 0.001 1 0.001 1 0.001 1 0.001 1 0.001 1 0.000 1 0.000 5 0.012 25 0.024 18 0.020 13 0.017 20 0.019 28 0.028 17 0.030 13 0.020

(MFs), or cellular components (CCs). The DoFD values measured by (7) and the similarity distribution over all gene pairs are reported in Table 2. The values in bracket mentioned after the name of each method depict the obtained gene pair numbers which have semantic similarity. All the top 50 genes selected by different gene selection algorithms are used in this analysis. In case of the MRMS algorithm, SVM and K-NN classifiers generate different sets of optimum values of β and η parameters. So, in gene ontology based analysis, results are presented for optimum values of β and η obtained by both SVM and K-NN. From Table 2, it can be observed that the similarity distribution of most gene pairs in the gene sets are located at the lower end for all datasets

484

S. Paul and P. Maji

and in three ontologies. In all ontologies and datasets, the DoFD values of genes of the MRMS method are higher than that obtained other gene selection algorithms.

4 Conclusion In this paper, an application of rough set has been demonstrated to select functionally diverse genes from microarray data. For two cancer and one arthritis microarray data sets, significantly better results are found for the rough set based method compared to existing gene selection methods. The rough set based algorithm generates more functionally diverse sets of genes, which also provide high classification accuracy. All the results reported in this paper demonstrate the feasibility and effectiveness of the rough set based method. Acknowledgement. The work was done when one of the authors, S. Paul, was a Senior Research Fellow of Council of Scientific and Industrial Research, Government of India.

References 1. Breiman, L.: Random forests. Machine Learning 45(1), 5–32 (2001) 2. Domany, E.: Cluster Analysis of Gene Expression Data. Journal of Statistical Physics 110(36), 1117–1139 (2003) 3. Du, Z., Li, L., Chen, C.F., Yu, P.S., Wang, J.Z.: G-sesame: Web tools for go-term-based gene similarity analysis and knowledge discovery. Nucleic Acids Research 37, W345–W349 (2009) 4. Duda, R.O., Hart, P.E., Stork, D.G.: Pattern Classification and Scene Analysis. John Wiley and Sons, New York (1999) 5. Hall, M.: Correlation-Based Feature Selection for Discrete and Numeric Class Machine Learning. In: Proceedings of the Seventeenth International Conference on Machine Learning, pp. 359–366 (2000) 6. Loennstedt, I., Speed, T.P.: Replicated microarray data. Statistica Sinica 12, 31–46 (2002) 7. Maji, P., Paul, S.: Rough set based maximum relevance-maximum significance criterion and gene selection from microarray data. International Journal of Approximate Reasoning 52(3), 408–426 (2011) 8. Pal, S.K., Mitra, S.: Neuro-Fuzzy Pattern Recognition: Methods in Soft Computing. Wiley, New York (1999) 9. Tusher, V., Tibshirani, R., Chu, G.: Significance analysis of microarrays applied to the ionizing radiation response. Proceedings of the National Academy of Sciences 98, 5116–5121 (2001) 10. Vapnik, V.: The Nature of Statistical Learning Theory. Springer, New York (1995)

Quality Evaluation Measures of Pixel - Level Image Fusion Using Fuzzy Logic Srinivasa Rao Dammavalam1, Seetha Maddala2, and M.H.M. Krishna Prasad 3 1

Department of Information Technology, VNRVJIET, Hyderabad, India 2 Department of CSE, GNITS, Hyderabad, India 3 Department of CSE, JNTU College of Engineering, Vizianagaram, India {dammavalam2,krishnaprasad.mhm}@gmail.com, [email protected],

Abstract. Image fusion is a technique to combine the registered images to increase the spatial resolution of acquired low detail multi-sensor images and preserving their spectral information. In fusing panchromatic and multispectral images, the objective is to reduce uncertainty and minimize redundancy in the output while maximizing relevant information. Different fusion methods provide different results for different applications, medical imaging, automatic target guidance system, remote sensing, machine vision, automatic change detection, and biometrics. In this paper, we utilize a fuzzy logic approach to fuse images from different sensors, in order to enhance visualization. The work here further explores the comparison between image fusion using wavelet transform and fuzzy logic approach along with performance/quality evaluation measures like image quality index, entropy, mutual information measure, root mean square error, peak signal to noise ratio, fusion factor, fusion symmetry and fusion index. Experimental results prove that the use of the proposed method can efficiently preserve the spectral information while improving the spatial resolution of the remote sensing images. Keywords: image fusion, panchromatic, multispectral, wavelet transform, fuzzy logic, mutual information measure, image quality index, fusion factor, fusion symmetry, fusion index, entropy.

1

Introduction

Image fusion is a process of integrating information obtained from various sensors and intelligent systems. It provides a single image containing complete information. The concept of Image fusion has been used in wide variety of applications like medical imaging, remote sensing, navigation aid, machine vision, automatic change detection, biometrics etc. Yi Zheng et al. [1] proposed a method, Multisensor image fusion for surveillance systems in which fuzzy logic approach utilized to fuse images from different sensors, in order to enhance visualization for surveillance. In [2] XuHong Yang et. al. proposed, urban remote image fusion using fuzzy rules to refine the B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 485–493, 2011. © Springer-Verlag Berlin Heidelberg 2011

486

S.R. Dammavalam, S. Maddala, and M.H.M. Krishna Prasad

resolution of urban multi-spectral images using the corresponding high-resolution panchromatic images. After the decomposition of two input images by wavelet transform three texture features are extracted and then a fuzzy fusion rule is used to merge wavelet coefficients from the two images according to the extracted features. Zhu Mengyu et al. [3] present a image fusion algorithm based on fuzzy logic and wavelet, aimed at the visible and infrared image fusion and address an algorithm based on the discrete wavelet transform and fuzzy logic. In [3] the technique created two fuzzy relations, and estimated the importance of every wavelet coefficient with fuzzy reasoning. In [4] Rahul Ranjan et al. give a Iterative Fuzzy and Neuro Fuzzy approach for fusing medical images and remote sensing images and found that the technique very useful in medical imaging and other areas, where quality of image is more important than the real time application. In [ 5] a new method is proposed for PixelLevel Multisensor image fusion based on Fuzzy Logic in which the membership function and fuzzy rules of the new algorithm is defined using the Fuzzy Inference System. Yang-Ping Wang et al. [6] proposed, a fuzzy radial basis function neural networks is used to perform auto-adaptive image fusion and in experiment multimodal medical image fusion based on gradient pyramid is performed for comparison. In [7] a novel method is proposed using combine framework of wavelet transform and fuzzy logic and it provides novel tradeoff solution between the spectral and spatial fidelity and preserves more detail spectral and spatial information. Bushra et al. [8] proposed a method, Pixel & Feature Level Multi-Resolution Image Fusion based on Fuzzy logic in which images are first segmented into regions with fuzzy clustering and are then fed into a fusion system, based on fuzzy if-then rules.

2

Image Fusion Using Wavelet Transform

Wavelet Transform is a type of signal representation that can give the frequency content of the signal at a particular instant of time. Wavelet analysis has advantages over traditional Fourier methods in analyzing physical situations where the signal contains discontinuities and sharp spikes. Li et al. [9] proposed a multi-sensor image fusion using the wavelet transform method that the wavelet transforms of the input images are appropriately combined, and the new image is obtained by taking the inverse wavelet transform of the fused wavelet coefficients. An area-based maximum selection rule and a consistency verification step are used for feature selection. In [10] Wavelet transforms provide a framework in which a signal is decomposed, with each level corresponding to a coarser resolution, or lower frequency band. There are two main groups of transforms, continuous and discrete. Discrete transforms are more commonly used and can be subdivided in various categories. Susmitha et al. [11] Proposed a novel architecture for wavelet based fusion of images from different sources using multiresolution wavelet transforms which applies pixel based maximum selection rule to low frequency approximations and filter mask based fusion to high frequency details of wavelet decomposition. The key feature of hybrid architecture is

Quality Evaluation Measures of Pixel – Level Image Fusion Using Fuzzy Logic

487

the combination of advantages of pixel and region based fusion in a single image which can help the development of sophisticated algorithms enhancing the edges and structural details. Krista et al. [12] give a comprehensive literature survey of wavelet transform theory and an overview of image fusion techniques, and the results from a number of wavelet-based image fusion schemes are compared.

3

Image Fusion Based on Fuzzy Logic

It is important to know that the set of images used in this algorithm are registered images. Through registration we find correspondence between images and ensured that spatial correspondence established, fusion makes sense [8]. 3.1

Fuzzy Logic in Image Processing

Fuzzy logic has three main stages. Image fuzzification, modification of membership values, image defuzzification. The coding of image data (fuzzification) and decoding of the results (defuzzification) are steps that make possible to process images with fuzzy techniques. The main power of fuzzy image processing is in the middle step (modification of membership values). After the image data are transformed from gray-level plane to the membership plane (fuzzification), appropriate fuzzy techniques modify the membership values. It can be a fuzzy clustering, a fuzzy rulebased, fuzzy integration approach and so on [13].

3.2

Steps Involved in Fuzzy Image Processing

The original image in the gray level plane is subjected to fuzzification and the modification of membership functions is carried out in the membership plane. The result is the output image obtained after the defuzzification process. The first step fuzzification is done as follows in both the source images. •

Both the images are compared with each other.

The second step Modification of memberships is done as follows: • • •

If fa(x ) = fb (x ) then Ff (x ) = 0 If fa(x ) > fb (x ) then Ff (x ) = 1 If fa(x ) < fb (x ) then Ff (x ) = -1

The third step defuzzification is done as follows: • • •

If Ff (x ) = 0 then Ff (x ) = fa(x ) or fb (x ) If Ff (x ) = 1 then Ff (x ) = fa(x ) If Ff (x ) = -1 then Ff (x ) = fb (x )

488

S.R. Dammavalam, S. Maddala, and M.H.M. Krishna Prasad

Membership functions and rules considered in the fuzzy system 1. 2. 3.

if (input1 is mf2) and (input2 is mf3) then (output1 is mf2) if (input1 is mf1) and (input2 is mf2 then (output1 is mf3) if (input1 is mf3) and (input2 is mf2) then (output1 is mf2)

Algorithm steps for pixel level image fusion using Fuzzy Logic approach [14]. • • • • • • • • •

•

4

Read first image in variable I1 and find its size (rows: rl, columns: c1). Read second image in variable I2 and find its size (rows:r2, columns: c2). Variables I1 2 and I2 are images in matrix form where each pixel value is in the range from 0-255. Use Gray Colormap. Compare rows and columns of both input images. If the two images are not of the same size, select the portion, which are of same size. Convert the images in column form which has C= rl*cl entries. Make a fis (Fuzzy) file, which has two input images. Decide number and type of membership functions for both the input images by tuning the membership functions. Input images in antecedent are resolved to a degree of membership ranging 0 to 255. Make rules for input images, which resolve the two antecedents to a single number from 0 to 255. For num=l to C in steps of one, apply fuzzification using the rules developed above on the corresponding pixel values of the input images which gives a fuzzy set represented by a membership function and results in output image in column format. Convert the column form to matrix form and display the fused image.

Evaluation Measures

Evaluation measures are used to evaluate the quality of the fused image. The fused images are evaluated, taking the following parameters into consideration. 4.1

Image Quality Index

IQI measures the similarity between two images (I1 & I2) and its value ranges from -1 to 1. IQI is equal to 1 if both images are identical. IQI measure is given by [15]

IQI

=

m

2 xy 2 m a m

ab

b 2

(1)

Where x and y denote the mean values of images I1 and I2 and

ma , mb and

m am

b

x

2

+ y m

2

2

a

+ m

mab denotes the variance of I1, I2 and covariance of I1 and I2.

b

2

2

Quality Evaluation Measures of Pixel – Level Image Fusion Using Fuzzy Logic

4.2

489

Fusion Factor

Given two images A and B, and their fused image F, the Fusion factor (FF) is illustrated as [16]:

FF

=

I

AF

+

I

BF

(2)

Where IAF and IBF are the MIM values between input images and fused image. A higher value of FF indicates that fused image contains moderately good amount of information present in both the images. However, a high value of FF does not imply that the information from both images is symmetrically fused. 4.3

Fusion Symmetry

Fusion symmetry (FS) is an indication of the degree of symmetry in the information content from both the images.

  I AF − 0.5  FS = abs   I AF + I BF

(3)

The quality of fusion technique depends on the degree of Fusion symmetry. Since FS is the symmetry factor, when the sensors are of good quality, FS should be as low as possible so that the fused image derives features from both input images. If any of the sensors is of low quality then it is better to maximize FS than minimizing it. 4.4

Fusion Index

This study proposes a parameter called Fusion index from the factors Fusion symmetry and Fusion factor. The fusion index (FI) is defined as

FI = I AF / I BF

(4)

Where IAF is the mutual information index between multispectral image and fused image and IBF is the mutual information index between panchromatic image and fused image. The quality of fusion technique depends on the degree of fusion index. 4.5

Mutual Information Measure

Mutual information measure furnishes the amount of information of one image in another. This gives the guidelines for selecting the best fusion method. Given two images M (i, j) and N (i, j) and MIM between them is defined as:

I MN =  PMN (x, y )log x, y

PMN ( x, y ) PM (x )PN ( y )

(5)

Where, PM (x) and PN (y) are the probability density functions in the individual images, and PMN (x, y) is joint probability density function.

490

4.6

S.R. Dammavalam, S. Maddala, and M.H.M. Krishna Prasad

Root Mean Square Error

The root mean square error (RMSE) measures the amount of change per pixel due to the processing. The RMSE between a reference image R and the fused image F is given by

RMSE =

4.7

1 M N  (R(i, j ) − F (i, j )) MN i =1 j =1

(6)

Peak Signal to Noise Ratio

Peak signal to noise ratio (PSNR) can be calculated by using the formula

 L2  PSNR = 20 log10    MSE 

(7)

Where MSE is the mean square error and L is the number of gray levels in the image. 4.8

Entropy

Entropy E, a scalar value representing the entropy of grayscale image. Entropy is a statistical measure of randomness that can be used to characterize the texture of the input image. Entropy is defined as

E = − sum( p * log 2 ( p))

(8)

Where p contains the histogram counts returned from imhist.

5

The Experiment and Analysis

In this section, we fused a panchromatic image and multispectral image using our algorithm. Case 1, Panchromatic and Multispectral images of the Hyderabad city, AP, INDIA are acquired from the IRS 1D LISS III sensor at 05:40:44, Case 2 and Case 3 images are acquired from http://imagefusion.org [17]. The proposed algorithm has been implemented using Matlab 7.0. It can be seen from the above table and the image results that the fuzzy logic approach are having much better results when compared with the conventional technique. Table 1 shows that fuzzy based fused approach has shown comparatively better IQI, MIM and Entropy through preserving more spectral information. Considerable differences are obtained through fuzzy logic with lower RMSE and higher values of Fusion Factor and PSNR. So therefore it is concluded from experimental results that fuzzy logic based image fusion schemes perform better than conventional wavelet transform.

Quality Evaluation Measures of Pixel – Level Image Fusion Using Fuzzy Logic

Case 1:

(a)

Case 2:

(b)

(e)

Case 3:

(f)

(i)

(j)

491

(c)

(d)

(g)

(h)

(k)

(l)

Fig. 1. Some example images (a), (b), (e), (f), (i) and (j): original input images; (c), (g) and (k): fused images by wavelet transform and (d), (h) and (l): fused images by fuzzy logic Table 1. The evaluation measures of image fusion based on wavelet transform and fuzzy logic

Method Wavelet Transform (Case 1) (Case 2) (Case 3)

IQI

FF

FS

FI

MIM

RMSE

PSNR

Entropy

0.9473 3.8629 0.0429 1.1879 1.7656 63.5529 11.3425 7.3828 0.8650 3.8832 0.0118 0.9538 1.9875 19.1999 22.4648 7.2339 0.5579 2.6841 0.2731 3.4074 2.0751 39.5475 16.1884 5.9807

Fuzzy Logic (Case 1) 0.9689 5.5687 0.2752 3.4475 4.3166 52.5301 13.7226 7.3445 (Case 2) 0.9955 8.8407 0.0598 1.2719 3.8914 17.8385 23.1036 7.2577 (Case 3) 0.9896 4.7589 0.4023 9.2320 4.2938 25.4703 20.0101 6.7300

492

6

S.R. Dammavalam, S. Maddala, and M.H.M. Krishna Prasad

Conclusions

In this paper, a pixel level image fusion using fuzzy logic approach for remote sensing and medical imaging has been presented. All the results obtained and discussed by this method are same scene. In order to evaluate the results and compare the methods, the assessment criteria, evaluation measures are employed. The experimental results clearly show that the introduction of the proposed image fusion using fuzzy logic gives a considerable improvement on the performance of the fusion system. It is hoped that the technique can be further extended to all types of images and to integrate valid evaluation metric of image fusion schemes. And automatic determinations of the percentage of overlapping among fuzzy sets and membership functions are also worthy of research. Acknowledgments. This paper is partially supported by the All India Council for Technical Education, New Delhi, India under Research Promotion Scheme (RPS) Grant No. 8023/RID/RPS-80/2010-11.

References 1. Yi, Z., Ping, Z.: Multisensor Image Fusion Using Fuzzy Logic for Surveillance Systems. In: IEEE Seventh International Conference on Fuzzy Systems and Discovery, Shanghai, pp. 588–592 (2010) 2. Yang, X.H., Huang, F.Z., Liu, G.: Urban Remote Image Fusion Using Fuzzy Rules. In: IEEE Proceedings of the Eighth International Conference on Machine Learning and Cybernetics, Baoding, pp. 101–109 (2009) 3. Mengyu, Z., Yuliang, Y.: A New image Fusion Algorithm Based on Fuzzy Logic. In: IEEE International Conference on Intelligent Computation Technology and Automation, Changsha, pp. 83–86 (2008) 4. Ranjan, R., Singh, H., Meitzler, T., Gerhart, G.R.: Iterative Image Fusion technique using Fuzzy and Neuro fuzzy Logic and Applications. In: IEEE Fuzzy Information Processing Society, Detroit, USA, pp. 706–710 (2005) 5. Zhao, L., Xu, B., Tang, W., Chen, Z.: A Pixel-Level Multisensor Image Fusion Algorithm based on Fuzzy Logic. In: Wang, L., Jin, Y. (eds.) FSKD 2005. LNCS (LNAI), vol. 3613, pp. 717–720. Springer, Heidelberg (2005) 6. Wang, Y.P., Dang, J.W., Li, Q., Li, S.: Multimodal Medical Image fusion using Fuzzy Radial Basis function Neural Networks. In: IEEE International Conference on Wavelet Analysis and Pattern Recognition, Beijing, pp. 778–782 (2007) 7. Tanish, Z., Ishit, M., Mukesh, Z.: Novel hybrid Multispectral Image Fusion Method using Fuzzy Logic. I. J. Computer Information Systems and Industrial Management Applications, 9–103 (2010) 8. Bushra, N.K., Anwar, M.M., Haroon, I.: Pixel & Feature Level Multi-Resolution Image Fusion based on Fuzzy Logic. In: ACM Proc. of the 6th WSEAS International Conference on Wavelet Analysis & Multirate Systems, Romania, pp. 88–91 (2006) 9. Li, H., Manjunath, B.S, Mitra, S.K.: Multi-Sensor Image Fusion Using the Wavelet Transform. In: IEEE International Conference on Image Processing, Austin, pp. 51–55 (1994)

Quality Evaluation Measures of Pixel – Level Image Fusion Using Fuzzy Logic

493

10. Kannan, K., Perumal, S.A., Arulmozhi, K.: Performance Comparision of various levels of Fusion of Multi-Focused Images using Wavelet Transform. I. J. Computer Applications (2010) ISSN 0975–8887 11. Susmitha, V., Pancham, S.: A Novel Architecture for Wavelet based Image Fusion. World Academy of Science, Engineering and Technology, 372–377 (2009) 12. Krista, A., Zhang, Y., Dare, P.: Wavelet based Image Fusion Techniques- An Introduction Review and Comparision. J. Photogrammetry & Remote Sensing, 249–263 (2007) 13. Maruthi, R., Sankarasubramanian, K.: Pixel Level Multifocus Image Fusion Based on Fuzzy Logic Approach. J. Information Technology 7(4), 168–171 (2008) 14. Thomas, M., David, B., Sohn, E.J., Kimberly, L., Darryl, B., Gulshecn, K., Harpreet, S., Samuel, E., Grmgory, S., Yelena, R., James, R.: Fuzzy Logic based Image Fusion Aerosense, Orlando (2002) 15. Mumtaz, A., Masjid, A.: Genetic Algorithms and its Applicatio to Image Fusion. In: IEEE International Conference on Emerging Technologies, Rawalpindi, pp. 6–10 (2008) 16. Seetha, M., MuraliKrishna, I.V., Deekshatulu, B.L.: Data Fusion Performance Analysis Based on Conventional and Wavelet Transform Techniques. In: IEEE Proc. Geoscience and Remote Sensing Symposium, Seoul, pp. 284–2845 (2005) 17. The Online Resource for Research in Image Fusion, http://www.imagefusion.org

Load Frequency Control: A Polar Fuzzy Approach Rahul Umrao1, D.K. Chaturvedi2, and O.P. Malik3 1 Department of Electrical Engineering Faculty of Engineering, Dayalbagh Educational Institute, Dayalbagh, Agra, 282005, India [email protected] 2 Department of Electrical Engineering Faculty of Engineering, Dayalbagh Educational Institute, Dayalbagh, Agra, 282005, India [email protected] 3 Department of Electrical and Computer Engineering, University of Calgary, 2500 University Drive, N.W., Calgary, AB, T2N 1N4, Canada [email protected]

Abstract. The Performance of a Fuzzy logic controller is limited by its large number of rules and if the rules are large then computation time and requirement of memory is large. This problem is compensated by using Polar Fuzzy logic controller. So a Polar Fuzzy logic controller is proposed for the load frequency control problem. The aim of the Polar Fuzzy Controller is to restore the frequency and tie-line power in a smooth way to its nominal value in the shortest possible time if any load disturbance is applied. System performance is examined and compared with a standard Fuzzy logic controller, and conventional controllers.

1 Introduction Today the electrical power system is very much large and made with interconnected control areas. Load Frequency Control (LFC) is an important problem in electrical power system to keep the system frequency and the inter-area tie line power as close as possible to the scheduled values for reliable electrical power with good quality [1]. By controlling the mechanical input power to the generators of plants, frequency can be maintained as scheduled value. Due to variation of load and disturbances of system, which comes any time, frequency is disturbed from its rated value. The design of power system is such that it should maintain the frequency and voltage of the network within tolerable limits [2, 3]. The main objective of the load frequency controller is to exert control of frequency and at the same time of real power exchange via out going lines [4]. Several strategies for LFC have been proposed. Although the conventional control techniques were used in majority of literature [5], several studies using novel and intelligent control techniques for LFC are also reported in the literature. These controllers have shown good results in load frequency control [6]. Application of a Polar Fuzzy controller to LFC is described in this paper. Its performance on a single area and a two area thermal system with and without reheat unit is described, and compared with the conventional controllers like B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 494–504, 2011. © Springer-Verlag Berlin Heidelberg 2011

Load Frequency Control: A Polar Fuzzy Approach

495

proportional integral (PI), proportional integral derivative (PID), and intelligent controller like a standard Fuzzy logic controller.

2 Conventional Controllers The PI controller is popular in the industry. The proportional gain provides stability and high frequency response. The integral term insures that the average error is driven to zero. Advantages of PI are; only two gains need to be tuned, there is no long-term error, and the method normally provides highly responsive system. The predominant weakness is that PI controllers often produce excessive overshoot to a step command [7]. Conventional PID controller is the most widely used in the industry [8-9]. The PID controller, a combination of PI and PD, can be easily adopted to correspond to different controlled plants but it cannot yield a good control performance if the controlled system is of high order and nonlinear. The PD control, as in the case of the lead compensator, improves the transient-response characteristics, improves system stability, and increases the system bandwidth, which implies fast rise time [10].

3 Fuzzy Logic Controller After 1991 fuzzy logic application is used for industrial tools. Fuzzy controllers are preferred over conventional controllers because: •

Higher robustness.

•

To develop a fuzzy controller is cheaper.

•

It is easy to understand because it is expressed in natural linguistic terms.

•

It is easy to learn how fuzzy controller operates and how to design.

In last two decades the fuzzy set theory is a new methodological tool for handling ambiguity and uncertainty. Load frequency control has the main goal to maintain the balance between production and consumption [11-12]. The power system is very complex in nature because of there are several variable conditions. So the fuzzy logic helps to develop robust and reliable controller for load frequency control problem [3-5, 10].

Domain Expert

Fuzzy Knowledge Base and inference

Fuzzification

Pre-

Defuzzification Post-

Power System Fig. 1. Functional Module of a Simple Fuzzy Controller

496

R. Umrao, D.K. Chaturvedi, and O.P. Malik

Functional module of a simple fuzzy logic controller (FLC) is shown in Fig. 1.These controllers are useful for solving a wide range of control problems [13]. Five triangular, input and output both, membership functions are taken for design of FLC. 3.1

Fuzzy Rules

In FLC, five rules are used. These are: I. II. III. IV.

V.

If input is Z then output is LN. If input is LN then output is LP. If input is LP then output is LP. If input is MP then output is Z. If input is MP then output is Z. Δώ

4 Polar Fuzzy Controller The working of Polar fuzzy controller (PFC) is based on polar fuzzy sets. One important difference between the standard fuzzy sets and polar fuzzy sets is that polar fuzzy sets are described in polar quardinate. These sets are defined on the universe of angles, hence are repeating shapes every 2π cycles. There are two states for controlling the LFC problem 1. 2.

Sector F Sector A

R

Sector E

θ

Δω

O

Sector B Sector D Sector C Min Axis Fig. 2. Six Sector Phase Plane

Deviation in system frequency (Δf) Rate of change of system frequency (Δ˙f)

and the calculation of output of PFC is based on angle [14]. In six sector phase plane diagram origin ‘O’ is the desired equilibrium point as shown in Fig. 2. The control action is such that the direction of generator state R always towards origin ‘O’. LN and LP are two fuzzy input membership functions which are define in degree and these two are complimentary to each other where LN stands for large negative and LP for large positive. These two are defined in the range of 45o to 405o as shown in Fig. 3. As described earlier, generation of output of PFC (UFLC) is depending upon the utilization of fuzzy input (angle). P and N are the two triangular membership functions, which are defined in range of -1 to +1 as shown in Fig. 4 where P stands for positive and N for negative.

Load Frequency Control: A Polar Fuzzy Approach

4.1

497

Working of PFC

The working of PFC is described in appendix.

Fig. 3. Fuzzy sets for output variable

Fig. 4. Fuzzy sets for output variable

5 Application of Polar Fuzzy Logic Controller for LFC The above developed PFC has been used for load frequency control of single area and two area thermal systems. All studies are for a disturbance of 1% step change in load in one or both areas. 5.1

Load Frequency Control of Single Area System

The quality of supply depends on the voltage and frequency variations. The frequency and the voltage at the generator terminals are determined by the mechani cal input and are also affected by the change in load/demand [15]. To control the variations in voltage and frequency, sophisticated controllers detect the load variation and control the input to counter balance the load variation. In conventional control ΔPd (s) ΔPc (s)

ΔF (s)

ࡷࢍ ࡷ࢚

ࡷ࢖࢙

൫૚ ൅ ࢙ࢀࢍ൯ሺ૚ ൅ ࢙ࢀ࢚ ሻ

൫૚ ൅ ࢙ࢀ࢖࢙ ൯ ૚ ࡾ

Fig. 5. Block diagram of single area thermal power plant with PI controller

498

R. Umrao, D.K. Chaturvedi, and O.P. Malik

systems, the integral part of the controller controls the frequency variation to zero as early as possible by getting the optimum value of the integral gain Ki. A small signal model of the single area system is shown in Fig. 5. The turbine output and the electrical load perturbation given in the generator block, gives ‘ΔF(s)’ or ‘df’ as the output. The single area thermal system described above has been simulated in Matlab 7.0/ Simulink environment. 0

0

-0.005

-0.01 -0.01

Fuzzy

-0.03

Fuzzy -0.025

PID

df

df

PID

-0.02

PI

-0.02

PI

-0.015

Polar Fuzzy

Polar Fuzzy

-0.04

-0.03 -0.035 0

5

10

15

20

25

30

-0.05 0

10

20

30

Time(Sec)

Time(Sec)

Fig. 6. Frequency variations of single area Fig. 7. Frequency variation of single area with without Reheat Thermal system Reheat Thermal system

Table 1. Performance comparison of different controllers without and with reheat unit when 1% disturbance in single area thermal system Settling Time(sec) Controller

Undershoot(Hz)

PI

Without Reheat Unit 28

With Reheat Unit 30

Without Reheat Unit -0.031

With Reheat Unit -0.046

PID

18

20

-0.030

-0.044

Fuzzy

13

15

-0.034

-0.049

Polar Fuzzy

10

12

-0.032

-0.043

5.2

Load Frequency Control of Two Area System

When two single areas are tied together through a tie line to maintain their frequency and power exchange as per area contract is called two area system and electric power is transferred through the tie line between the areas [13]. Frequency will deviates from its steady state value if demand in any area is changed. The Simulink model of the two area system has been developed to study the system under different perturbations. With the help of this model system performance can be checked for different controllers for different perturbations. Performance of the PFC has been compared with standard FLC and conventional controllers for with and without reheat unit in the two area thermal system. The results are shown in Figs. 8-10 and Figs. 11-13, respectively, and in Tables 2 and 3.

Load Frequency Control: A Polar Fuzzy Approach

499

0

0.005 0

-0.005 PI

-0.01

PI

df2

df1

-0.005

PID

-0.015

PID

-0.01

Fuzzy

Fuzzy

Polar Fuzzy

Polar Fuzzy

-0.02

-0.015

-0.025 -0.03 0

10

20 30 Time(Sec)

40

-0.02 0

50

10

20 30 Time(Sec)

40

50

Fig. 8. Frequency variation of area- 1in Two Fig. 9. Frequency variation of area- 2 in a Two Area Thermal System without Reheat when Area Thermal System without Reheat when disturbance in area – 1 the disturbance in area – 1

0.01

0.01

0

0

-0.01

-0.01

PID

-0.02

PI

df1

df1

PI

PID

-0.02

Fuzzy

Fuzzy

Polar Fuzzy

Polar Fuzzy

-0.03 -0.04 0

-0.03

10

20 30 Time(Sec)

40

-0.04 0

50

10

20 30 Time(Sec)

40

50

0.01

0

0

-0.005

-0.01

-0.01

-0.02

PI PID Fuzzy Polar Fuzzy

-0.03 -0.04 -0.05 0

df2

df1

Fig. 10. Frequency variation of area- 1 in a Fig. 11. Frequency variation of area- 1 in a Two Area Thermal System without Reheat Two Area Thermal System without Reheat when disturbance in both areas when disturbance in area - 1

10

20 30 Time(Sec)

40

PI PID

-0.015

Fuzzy

-0.02

Polar Fuzzy

-0.025 50

-0.03 0

10

20 30 Time(Sec)

40

50

Fig. 12. Frequency variation of area- 2 in a Fig. 13. Frequency variation of area- 1 in a Two Area Thermal System with Reheat unit Two Area Thermal System with Reheat unit when disturbance in area - 1 when disturbance in both areas

500

R. Umrao, D.K. Chaturvedi, and O.P. Malik

Table 2. Performance comparison of different controllers in two area thermal system without reheat unit PI 1% Distur- Resp. of bance Area area

PID

Settling Under shoot Time (Hz) (Sec)

Settling Under shoot Time (Hz) (Sec)

Fuzzy Settling Time (Sec)

Under shoot (Hz)

Polar Fuzzy Settling Under shoot Time (Hz) (Sec)

Area 1

Area 1

26

-0.029

20

-0.025

12

-0.025

7

-0.023

Area 1

Area 2

32

-0.015

25

-0.013

15

-0.013

10

-0.010

Both Areas Area 1

28

-0.035

22

-0.030

13

-0.030

8

-0.027

Both Areas Area2

28

-0.031

22

-0.030

13

-0.030

8

-0.027

Table 3. Performance comparison of different controllers in two area thermal system with reheat unit PI

1% Distur- Resp. of bance Area area

PID

Settling Under shoot Time (Hz) (Sec)

Settling Under shoot Time (Hz) (Sec)

Fuzzy

Settling Time (Sec)

Under shoot (Hz)

Polar Fuzzy

Settling Under shoot Time (Hz) (Sec)

Area 1

Area 1

43

-0.032

35

-0.032

20

-0.032

13

-0.030

Area 2

Area 2

46

-0.027

35

-0.026

21

-0.027

14

-0.022

Both Areas Area 1

43

-0.046

35

-0.044

20

-0.045

13

-0.040

Both Areas Area 2

45

-0.046

35

-0.044

20

-0.045

13

-0.040

6 CO-ordination OF PFC and Conventional Controllers of Multi-Area System The PFC can’t replace all other conventional controllers in a time. This replacement is time taken process over all plants. So it is necessary to study the effect of PFC coordination with other conventional controllers. Most of the controllers used in the electrical power system are PI type. Therefore, the effect of the conventional PI and Polar Fuzzy controllers working together needs to be investigated. The comparative performance is studied when the conventional PI is installed in area #1 and the proposed PFC in area #2. Result is compared in Fig. 14 and Tables 4. It can be seen that the performance in terms of settling time and undershoots is much better when PFC is

Load Frequency Control: A Polar Fuzzy Approach 0.01 0 -0.01 df1

present than with the PI trollers only. The above studies show that the system response, when PFC is present, settles to zero steady state error in less time, with least oscillation and with minimum undershoot compared to the other conventional and Fuzzy logic controllers.

501

-0.02 Both PI

-0.03 PI in Area # 1 and PFC in Area # 2

-0.04 -0.05 0

10

20 30 Time(Sec)

40

50

Fig. 14. Frequency variation of area- 1 in Two Area Thermal System with Reheat unit when disturbance in both areas

Table 4. Performance comparison of "Both PI" and co-ordination of "PI with Polar Fuzzy controller" in two area thermal system with reheat

Both PI

Disturbance

Response of Area

1% disturbance in Area 1 1% disturbance in Area 2 1% disturbance in both Areas

Settling Time(Sec)

Co-ordination of PI with Polar fuzzy

Undershoot (Hz)

Settling Time(Sec)

Undershoot (Hz)

Area 1

43

-0.032

15

-0.032

Area 2

46

-0.027

30

-0.025

Area 1

45

-0.028

25

-0.024

Area 2

45

-0.032

25

-0.030

Area 1

43

-0.046

20

-0.042

Area 2

45

-0.046

20

-0.040

7 Conclusions A Polar Fuzzy controller is designed for automatic load frequency control of single area and two area interconnected thermal power systems with and without reheat unit. The controllers performance is compared based on the system settling time and undershoot. A comparison of the system dynamic response with the various controllers shows that the Polar Fuzzy controller yields improved control performance when compared to the PI and PID conventional and standard fuzzy logic controllers. It is also seen that the PFC performed better than the other conventional PI controller in the comparison study of coordination of PFC and conventional PI controller.

502

R. Umrao, D.K. Chaturvedi, and O.P. Malik

References [1] Chaturvedi, D.K., Satsangi, P.S., Kalra, P.K.: Load frequency control: a generalized neural network approach. Int. J. on Electric Power and Energy Systems 21(6), 405–415 (1999) [2] Chaturvedi, D.K.: Electrical Machines Lab Manual with Matlab Programs. University Science Press, New Delhi (2010) [3] Moon, Y.H., Ryu, H.S., Kim, B., Kim, S., Park, S.C.: Fuzzy logic based extended integral control for load frequency con. IEEE Power Engineering Society Winter Meeting 3, 1289–1293 (2001) [4] Anand, B., Jeyakumar, A.E.: Load frequency control with fuzzy logic controller considering non-linearities and boiler dyna. ICGST-ACSE Journal 8, 15–20 (2009) [5] Mathur, H.D., Manjunath, H.V.: Study of dynamic performance of thermal units with asynchronous tie-lines using fuzzy based cont. Journal of Electrical System 3, 124–130 (2007) [6] Chaturvedi, D.K.: Modeling and Simulation of Systems Using Matlab@/Simulink@. CRC Press, New York (2009) [7] Das, D., Kothari, M.L., Kothari, D.P., Nanda, J.: Variable structure control strategy to automatic generation control of interconnected reheat thermal systems. Proc. Inst. Elect. Eng. Contr. Theory App. 138(6), 579–585 (1991) [8] Ogata, K.: Modern control engineering. PHI Publication (2002) [9] Moon, Y.H., Ryu, H.S., Choi, B.K., Cho, B.H.: Modified PID load frequency control with the consideration of valve position limits. IEEE Power Engineering Society Winter Meeting 1, 701–706 (1999) [10] Nanda, J., Mangla, A.: Automatic generation control of an interconnected hydro-thermal system using conventional integral and fuzzy logic controller. In: IEEE International Conference on Electric Utility Deregulation, Restructuring and Power Technologies, vol. 1, pp. 372–377 (2004) [11] Ibraheem, K.P., Kothari, D.P.: Recent philosophies of automatic generation control strategies in power systems. IEEE Transactions on Power Systems 20(1), 346–357 (2005) [12] Chaturvedi, D.K.: Soft Computing: Applications to electrical engineering problem. Springer, Berlin (2008) [13] Kundur, P.: Power system stability and control. TMH Publication (2007) [14] Chaturvedi, D.K., Malik, O.P., Choudhury, U.K.: Polar fuzzy adaptive power system stabilizer. J. of The Institution of Engineers 90, 35–45 (2009) [15] Nagrath, I.J., Kothari, D.P.: Power System Engineering. TMH Publication (2005)

Appendix: Working of PFC It’s working is described in Chaturvedi (2009) [14]. In Polar Fuzzy logic based controller, there is no need to use two separate input gains for Δf and Δ˙f, because only one input, the polar angle, that depends on the ratio of the properly scaled inputs is used. Thus, only one gain, Kacc, is considered. The scaling factor Kacc decides as to which variable, frequency deviation or rate of change of frequency has more weight in the magnitude R. In the proposed controller the magnitude of the FLC output is set to be maximum at 450, because at this instant the directions of Δ f and Δ˙f are the same and magnitude is also high. On the other hand at 1350 axis the directions of Δ f and Δ˙f are opposite and, therefore, minimum control action is required. The maximum and minimum is fixed at these angles. But due to the scaling of Δ˙f with the gain

Load Frequency Control: A Polar Fuzzy Approach

503

Kacc, all the points in the phase plane are relocated and sometimes system conditions may also require these points to be relocated. Hence, for better tuning of the controller, there is a need for clockwise or anticlockwise rotation. This can be done by adding or subtracting an angle ‘β’ from phase plane angle ‘θ’ of the polar form. The polar fuzzy logic controller, shown in Fig. 5, uses angle θ’ as input which is defined as: θ’ =(θ – β) + 360o for θ – β < 45o θ’ =θ – β

for θ – β ≥ 45o.

The required control strategy is [refer Fig 2] a) In sector A (0 o – 90 o) control from FLC should be large positive as both scaled Δf and Δ˙f are positive. b) In Sector B (315 o - 360 o) control signal from FLC should be low positive as scaled Δf is large positive and scaled Δ˙f is small negative. c) In sector C (270 o - 315 o) control signal from FLC should be low negative as scaled Δf is small positive and scaled Δ˙f is large negative. In sectors D, E, F all the situations are completely opposite to those in sector A, B and C, respectively. At an angle (θ’) of 45o or 405o, the value of membership function of LP is maximum and that for LN is minimum so that ‘UFLC’ is positive maximum. At angles of 135o and 315o the value of membership function for both LP and LN is the same, so that UFLC is minimum (zero). At an angle of 2250 the value of membership function LP is minimum and that for LN is maximum so that UFLC is negative maximum. The working of PFC controller is shown in Fig. 15. Output of the fuzzy logic controller is divided into two linguistic variables ‘P’ and ‘N’, which are triangular membership functions. So here only two simple rules are considered. d) Rule 1 - If θ’ is LP then UFLC is P. Rule 2 - If θ’ is LN then UFLC is N. Hence, the output of FLC unit is UFLC=f1( θ, β), and final output u= UFLC* R+ Ko*Δf

Where, f1 – is a non-linear fuzzy functional mapping; θ – angle in degree; R – Magnitude; β – modifier (tuning parameter) in degree; Ko – multiplier (tuning parameter); Δf and Δ˙f – system frequency and rate of change of system frequency

504

R. Umrao, D.K. Chaturvedi, and O.P. Malik

Δf

Δ˙f

β

Kacc

_ Coordinate converter

+ θ

∑ No

R

(θ-β)>45o

360o +

Ko Π + +

∑

+

Yes

+ ∑

∑ + FLC

UFLC

θ’

u Fig. 15. Block Diagram of Proposed Polar Fuzzy Logic Controller

An Efficient Algorithm to Computing Max-Min Post-inverse Fuzzy Relation for Abductive Reasoning Sumantra Chakraborty1, Amit Konar1, and Ramadoss Janarthanan2 1 Artificial Intelligence Laboratory, Dept. of Electronics and Tele-Communication Engineering, Jadavpur University, Calcutta- 32, India 2 Department of IT, Jaya Engg. College, Chennai, India [email protected], [email protected], [email protected]

Abstract. This paper provides an alternative formulation to computing the maxmin post-inverse fuzzy relation by minimizing a heuristic (objective) function to satisfy the inherent constraints of the problem. An algorithm for computing the max-min post-inverse fuzzy relation as well as the trace of the algorithm is proposed here. The algorithm exposes its relatively better computational accuracy and higher speed in comparison to the existing technique for post-inverse computation. The betterment of computational accuracy of the max-min post-inverse fuzzy relation leads more accurate result of fuzzy abductive reasoning, because, max-min post-inverse fuzzy relation is required for abductive reasoning. Keywords: Max–min inverse fuzzy relation, Heuristic Function, Abductive reasoning.

1

Introduction

A fuzzy relation R(x, y) usually describes a mapping from universe X to universe Y (i.e. X →Y), and is formally represented by R(x, y) = {((x, y), μ R ( x, y ) )⏐(x, y) ∈ X × Y},

(1)

where, μ R ( x, y ) denotes the membership of (x, y) to belong to the fuzzy relation R(x, y). 1.1

Fuzzy Max-Min Post-inverse Relation

Let X, Y and Z be three universes and R1 (x, y), for (x, y) ∈ X × Y and R 2 (y, z), for

(y, z) ∈ Y × Z be two fuzzy relations. Then max-min composition operation of R1 and R2 , denoted by R1 o R 2 , is a fuzzy relation defined by B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 505–519, 2011. © Springer-Verlag Berlin Heidelberg 2011

506

S. Chakraborty, A. Konar, and R. Janarthanan

R1 o R2 = {(x, z),

max {min{ μ R1 (x, y), μ R2 (y, z)}}}

(2)

y

where x ∈ X, y ∈ Y and z ∈ Z. For brevity we would use ‘∧’ and ‘∨’ to denote min and max respectively. Thus, R1 o R 2 = {(x, z),

∨y

{ μ R1 (x, y) ∧ μ R2 (y, z)}}.

(3)

Let R1 and R 2 be two fuzzy relational matrices of dimension (n × m) and (m × n) respectively. When R1 o R 2 = I, the identity relation, we define R 2 as the post-inverse to R1 . It is easy to note that when R1 = R 2 = I, R1 o R 2 = I follows. However, when R1 ≠ I, we cannot find any R 2 that satisfies R1 o R 2 = I. It is apparent from the last statements that we can only evaluate approximate max-min post-inverse to R1 , when we know R1 . 1.2

Review

The origin of the proposed max-min fuzzy inverse computation problem dates back to the middle of 1970’s, when the researchers took active interest to find a general solution to fuzzy relational equations involving max-min composition operation. The pioneering contribution of solving max-min composition-based relational equation goes to Sanchez [36]. The work was later studied and extended by Prevot [31], Czogala, Drewniak and Pedrycz [8], Lettieri and Liguori [20], Luoh et al. [22], and Yeh [41] for finite fuzzy sets [16]. Cheng-Zhong [7] and Wang et al. [40] proposed two distinct approaches to computing intervals of solutions for each element of an inverse fuzzy relation. Higashi and Klir in [14] introduced a new approach to computing maximal and minimal solutions to a fuzzy relational equation. Among the other wellknown approaches to solve fuzzy relational equations, the works presented in [10], [12], [13], [15], [29], [43], [44], [30], [21] need special mention. The early approaches mentioned above unfortunately is not directly applicable to find general solutions R 2 , satisfying the relational equation: R1 o R 2 = I. Interestingly, there are problems like fuzzy backward/abductive [17] reasoning, where R1 ≠ I, but R 2 needs to be evaluated. This demands a formulation to determine a suitable R 2 , such that the given relational equation is best satisfied. Several direct (or indirect) formulations of the max-min pre-inverse computing problem have been addressed in the literature [2], [6], [7], [24], [25], [26], [27], [34], [35], [37], [45]. A first attempt to compute fuzzy pre-inverse with an aim to satisfy all the underlying constraints in the relational equation using a heuristic objective function is addressed in [35]. The work, however, is not free from limitations as the motivations to optimize the heuristic (objective) function to optimally satisfy all the constraints are not fully realized. All the limitations in [35] are fully resolved by [45].

An Efficient Algorithm to Computing Max-Min Post-inverse Fuzzy Relation

507

This paper is an extended part of [45], where the proposed work illustrates an alternative formulation to computing the max-min post-inverse fuzzy relation. The rest of the paper is organized as follows. In section 2, we provide Strategies used to solve the post-inverse computational problem. The algorithm is presented in section 3. The analysis of algorithm and example is given section 3.1 and 3.2 respectively.

2

Proposed Computational and Approach to Fuzzy Max-Min Post-inverse Relation

Given a fuzzy relational matrix R of dimension (m × n), we need to evaluate a Q matrix of dimension (n × m) such that R o Q = I′ ≈ I, where I denotes identity matrix of dimension (m × m). Let q j be the jth column of Q matrix. The following strategies have been adopted to solve the equation R o Q = I′ ≈ I for known R. Strategy 1: Decomposition of R o Q ≈ I into [R o q j ] j ≈ 1 and [R o q j ]l ,l ≠ j ≈ 0.

Since, R o Q ≈ I, R o q j ≈ jth column of I matrix, therefore, the jth element of R o q j , denoted by [R o q j ] j ≈ 1 and the lth element (where l ≠ j) of R o q j , denoted by [R o q j ]l ,l ≠ j ≈ 0. Strategy 2: Determination of the effective range of qij , ∀ i in [0, r ji ]. Since Q is a fuzzy relational matrix, its elements qij ∈ [0, 1] for ∀i, j. However, to satisfy the constraint [R o q j ] j ≈ 1, the range of qij , ∀i virtually becomes [0, r ji ] by Lemma 1. This range is hereafter referred to as effective range of qij . Lemma 1: The constraint [R o q j ] j ≈ 1, sets the effective range of qij in [0, r ji ]. Proof: [R o q j ] j

n

=

∨ ( r ji ∧ qij )

(4)

i =1 n

Since [

∨

i =1

( r ji ∧ qij )] q > r = [ ij ji

n

∨ ( r ji ∧ qij )] q

i =1

ij = r ji

,

508

S. Chakraborty, A. Konar, and R. Janarthanan

The minimum value of qij that maximizes [R o q j ] j towards 1 is r ji . Setting qij beyond r ji is of no use in connection with maximization of [R o q j ] j towards 1. Therefore, the effective range of qij reduces from [0, 1] to [0, r ji ].

Strategy 3: Replacement of the constraint [R o q j ] j ≈ 1, by qkj ≈ 1, where qkj ≥ qij ,∀i We first prove [R o q j ] j = qkj for qkj ≥ qij , ∀i by Lemma 2, and then argue that [R o q j ] j ≈ 1 can be replaced by qkj ≈ 1.

Lemma 2: If qkj ≥ qij ,∀ i, then [R o q j ] j = qkj . n

Proof: [R o q j ] j =

∨ ( r ji ∧ qij )

(5)

i =1

By Lemma 1, we can write 0 ≤ qij ≤ r ji , ∀i. Therefore, ( r ji ∧ qij ) = qij

(6)

Substituting expression (6) in expression (5), yields the resulting expression as

n

[R o q j ] j = = qkj

∨ ( qij )

(7)

i =1

as qkj ≥ qij , ∀i.

(8)

The maximization of [R o q j ] j , therefore, depends only on qkj , and the maximum value of [R o q j ] j = qkj . Consequently, the constraint [R o q j ] j ≈ 1 is replaced by

qkj ≈ 1. Discussion on Strategy 3 ends here. A brief justification to strategy 4-6 is outlined next.

Justification of Strategies 4 to 6: In this paper, we evaluate the largest element qkj and other element qij (for i ≠ k) in q j , the jth column of Q-matrix, by separate procedures. For evaluation of qkj , we first need to identify the positional index k of qkj so that maximization of [R o q j ] j and minimization of [R o q j ]l ,l ≠ j occur jointly for a suitable selection of qkj . This is taken care of in Strategy 5 and 6. In Strategy 5, we

An Efficient Algorithm to Computing Max-Min Post-inverse Fuzzy Relation

509

determine k for the possible largest element qkj , whereas in Strategy 6 we evaluate qkj . To determine qij (for i ≠ k), we only need to minimize [R o q j ]l ,l ≠ j . This is considered in Strategy 4. It is indeed important to note that selection of qij (for i ≠ k) to minimize [R o q j ]l ,l ≠ j does not hamper maximization of [R o q j ] j as [R o

q j ] j = qik , vide Lemma 2. Strategy 4: Evaluation of qij , i ≠ k, where qkj ≥ qij , ∀ i. The details of the above strategy are taken up in Theorem 1.

Theorem 1: If qkj ≥ qij ,∀i, then the largest value of qij that minimizes [R o i≠k m q j ]l ,l ≠ j towards 0 is given by ( rlk )∧ qkj . l =1 l≠ j

∧

Proof: [R o q j ]l ,l ≠ j n

=

∨ ( rli ∧ qij ),

i =1

∀ l, l ≠ j

n

=

∨ ( rli ∧ qij ) ∨ ( rlk ∧ qkj ),

i =1 i≠k

(9)

∀ l, l ≠ j

(10)

= ( rlk ∧ qkj ) ∀ l, l ≠ j, if ( rlk ∧ qkj ) ≥

n

∨ ( rli ∧ qij ).

i =1 i ≠k

(11)

Therefore, Min [R o q j ]l ,l ≠ j = =

Min ( rlk ∧ qkj ) ∀l , l ≠ j

Min { rlk } ∧ qkj ∀l , l ≠ j m

=(

rlk )∧ qkj ∧ l =1 l≠ j

(12)

510

S. Chakraborty, A. Konar, and R. Janarthanan

m

∧

rlk )∧ qkj and the largest value in [R o q j ]l ,l ≠ j = l =1 l≠ j ( rlk ∧ qkj ), therefore, Min[R o q j ]l ,l ≠ j , will be the largest among ( rli ∧ qij ) ∀ i, i ≠ k, if m rlk )∧ qkj Min[R o q j ]l ,l ≠ j = ( l =1 l≠ j Since Min[R o q j ]l ,l ≠ j = (

∧

n

≥

m

which is same as, (

∧ rlk )∧ qkj

l =1 l≠ j

∨ ( rli ∧ qij ),

i =1 i≠k

≥ ( rli ∧ qij )

∀ i, i ≠ k.

(13)

(14)

The largest value of qij for i ≠ k can be obtained by setting equality in equation (14), and the resulting equality condition is satisfied when m rlk )∧ qkj . qij =( i≠k l =1 l≠ j

∧

(15)

Strategy 5: Determining the positional index k for the element qkj ( ≥ qij , ∀j ) in qj . To determine the position k of qkj in q j , we first need to construct a heuristic function h( qkj ) that satisfies two constraints: i)

Maximize [R o q j ] j

(16)

ii)

Minimize [R o q j ]l ,l ≠ j ,

(17)

and then determine the index k, such that h( qkj ) ≥ h( qij ), ∀i. In other words, we need to determine the positional index k for the possible largest element qkj in the jth

An Efficient Algorithm to Computing Max-Min Post-inverse Fuzzy Relation

511

column of Q-matrix, as the largest value of h( qkj ) ensures maximization of the heuristic function h( qkj ), and thus best satisfies the constraints (16) and (17). Formulation of the heuristic function is considered first, and the determination of k satisfying h( qkj ) ≥ h( qij ), ∀i is undertaken next. One simple heuristic cost function that satisfies (16) and (17) is h1 ( qkj ) = qkj −

1 ( m − 1)

m



l =1 l≠ j

( rlk ∧ qkj ),

where qkj ≥ qij , ∀i, by Theorem 2. Next we find k such that Max Max h1 ( qkj ) ≥ h1 ( qij ), for all i. qkj ∈[0, r jk ] qij ∈[0,r ji ] Theorem 2: If qkj ≥ qij , ∀i, then maximization of [R o q j ] j and minimization of [R o q j ]l ,l ≠ j can be represented by a heuristic function, h1 ( qkj ) = qkj −

m 1 ( r ∧ q ). ( m − 1) l =1 lk kj l≠ j



Proof: Given qkj ≥ qij , for all i, Thus from Lemma 2, we have [R o q j ] j = qkj

(18)

= ( rl1 ∧ q1 j )∨( rl 2 ∧ q2 j )∨…∨( rlk ∧ qkj )∨...∨ ( rln ∧ qnj ), for ∀l , l ≠ j.

(19)

Further, [R o q j ]l ,l ≠ j

m =( Now by Theorem 1 we have qij rlk )∧ qkj and substituting this value in i≠k l =1 l≠ j equation (19) we have

∧

[R o q j ]l ,l ≠ j = ( rlk ∧ qkj ), for ∀l, l ≠ j

(20)

Now to jointly satisfy maximization of [R o q j ] j and minimization of [R o q j ]l ,l ≠ j we design a heuristic function,

512

S. Chakraborty, A. Konar, and R. Janarthanan

h1 ( qkj ) = [R o q j ] j − f(( r1k ∧ qkj ),…,( r j −1,k ∧ qkj ), ( r j +1, k ∧ qkj ), ..…,

( rmk ∧ qkj )). Now, for any monotonically non-decreasing function f(·), maximization of [R o q j ] j and minimization of the min terms ( r1k ∧ qkj ), ….…, ( r j −1,k ∧ qkj ), ( r j +1,k ∧ qkj ), ....…, ( rmk ∧ qkj ) calls for maximization of h1 ( qkj ). Since averaging (Avg) is a monotonically increasing function, we replace f(·) by Avg(·). Thus, h1 ( qkj ) = [R o q j ] j − Avg(( r1k ∧ qkj ),…,( r j −1,k ∧ qkj ), ( r j +1,k ∧ qkj ),

...…,( rmk ∧ qkj )) m 1 = qkj − ( r ∧ q ). ( m − 1) l =1 lk kj l≠ j



(21)

Although apparent, it may be added for the sake of completeness that n>1 in (21). The determination of index k, such that h1 ( qkj ) ≥ h1 ( qij ), ∀i can be explored now. Since qij ∈ [0, r ji ] and qkj ∈ [0, r jk ], therefore h1 ( qkj ) ≥ h1 ( qij ), ∀i can be transformed to h1 ( qkj ) ≥ h1 ( qij ), for all i Max Max qkj ∈[0, r jk ] qij ∈[0, r ji ]

(22)

Consequently, determination of k satisfying the inequality (22) yields the largest element qkj in the q j , that maximizes the heuristic function h1 (.). Corollary 1: h2 ( qkj ) = qkj −

m



( rlk ∧ qkj ), too is a heuristic function tha l =1 l≠ j t maximizes [R o q j ] j and minimizes [R o q j ]l ,l ≠ j . m

Proof: Since



( rlk ∧ qkj ) is a monotonically non-decreasing function, thus the

l =1 l≠ j Corollary can be proved similar to Theorem 2.

An Efficient Algorithm to Computing Max-Min Post-inverse Fuzzy Relation

513

Strategy 6: Finding the maximum value of h1 ( qij ) for qij in [0, r ji ]. We first of all prove that h1 ( qij ) is a monotonically non-decreasing function of qij

by Theorem 3. Then we can easily verify that for qij in [0, r ji ], h1 ( qij ) is qij = r ji the largest, i.e., h1 ( qij )

qij = r ji

≥ h1 ( qij )

qij ∈ [0, r ji ]

m 1 ( m − 1) l =1 l≠ j decreasing function of qij in [0, r ji ].



Theorem 3: h1 ( qij ) = qij −

.

( rli ∧ qij ), is a monotonically non-

Proof: We consider two possible cases:

Case 1: If qij > rli ∀ l, l ≠ j, then, h1 ′( qij ) =

dh1 (qij ) dqij

m

=1−



d (rli ) 1 m 1 − ( ) l =1 dqij l ≠i

= 1 (> 0).

(since,

d (rli ) = 0 ). dqij

Case 2: Let qij ≤ rli for at least one l (say t times), then, dh1 (qij ) h1 ′( qij ) = dqij =1−

=1−

d 1 ( m − 1) dqij



t −times

( qij )

t ≥ 0 as t ≤ (m−1). ( m − 1)

∴ h1 ′( qij ) ≥ 0, and therefore, h1 (·) is a monotonically non-decreasing function of qij in [0, r ji ]. Strategy 7: Finding Q-matrix Evaluation of each row q j of Q-matrix is considered independently. For given row q j , we need to determine the largest element qkj ≥ qij , ∀i. After qkj is evaluated, we evaluate qij , i ≠ k in the subsequent phase. Since maximization of h1 (·) ensures satisfaction of the constraints (16) and (17) in strategy 5, to determine qkj , we look for an index k, such that

514

S. Chakraborty, A. Konar, and R. Janarthanan

Max Max h1 ( qkj ) ≥ h1 ( qij ), for all i. qkj ∈[0, r jk ] qij ∈[0,r ji ] Further, since h1 (·) is a monotonically non-decreasing function, above inequality reduces to, h1 ( qkj )

qkj = r jk

≥ h1 ( qij )

qij = r ji

(23)

If a suitable value of k is found satisfying (23), we say qik is the largest element in qi and the value of qkj = r jk . For other element qij in q j , i ≠ k, we evaluate m rlk )∧ qkj qij =( i≠k l =1 l≠ j

∧

m

=(

∧ rlk )∧ r jk

(Since, qkj = r jk )

l =1 l≠ j

m

=

∧

rlk . (24) l =1 The principle for evaluation of qij for a given j can be repeated for j = 1 to m to determine Q-matrix.

3

Proposed Fuzzy Max-Min Post-inverse Computation Algorithm

The results obtained from the strategies in Section 2 are used here to construct Algorithm for Max-Min Post-inverse computation for a given fuzzy relational matrix. ALGORITHM 1 Input: R = [ r ] m×n where 0 ≤ r ≤ 1 ∀ i,j ;

ij ij Output: Q = [ q ] n×m where 0 ≤ q ≤ 1 ∀ i, j; ij ij //such that RoQ is close enough to I// Begin For j = 1 to m

An Efficient Algorithm to Computing Max-Min Post-inverse Fuzzy Relation

Evaluate qkj (.);

515

//Determine the position k and value of the largest //element qkj in column j of Q.//

For i = 1 to n If( i ≠ k ) Then qij = Min { rlk }; l //Determine all the elements in the jth column of Q matrix //except qkj // End If; End For; End For; End.

Evaluate qkj (.) Begin For i = 1 to n

h1 ( qij ) = r ji −

1 m ( − 1)

m



l =1 l≠ j

( rli ∧ r ji );

//Evaluate h1 ( qij ). //

End For If h1 ( qkj ) ≥ h1 ( qij ) for all i Then return k and qkj = r jk ; End. 3.1

//Return the position k of qkj , and its value//

Explanation of Algorithm I

Algorithm 1 evaluates the elements in q j , i.e., q1 j , q2 j ,……., qmj in a single pass by determining the position k of the largest element qkj in the jth column and then its value r jk . Next, we determine the other elements in q j , which is given by m = qij rlk . i≠k l =1

∧

The outer For-loop in the algorithm sets j = 1 to m with an increment in j by 1, and evaluation of q j takes place for each setting of j. The most important step inside this outer For-loop is determining positional index k of the largest element qkj in q j and evaluation of its value. This has been taken care of in function Evaluate qkj (.).

516

S. Chakraborty, A. Konar, and R. Janarthanan Table 1. Trace of the algorithm

j =1 Evaluate qkj (j) When q11 = r11 =0.9; h( q11 ) = When q21 = r12 =0.4; h( q21 ) =

r11 -{( r21 ∧ r11 )+( r31 ∧ r11 )}/2 = 0.8 r12 -{( r22 ∧ r12 )+( r32 ∧ r12 )}/2 = 0.05

When q31 = r13 =0.3; h( q31 ) = r13 -{( r23 ∧ r13 )+( r33 ∧ r13 )}/2 = 0.05 Max{ h( q11 ) , h( q21 ) , h( q31 ) }= h( q11 ) ; Return q11 =0.9 and k=1 Evaluate the largest qij for all i except k =1

q21 =Min( r11 , r21 , r31 ) = 0.1 ; q31 =Min( r11 , r21 , r31 ) = 0.1 ; j =2 Evaluate_ qkj (j) When q12 = r21 =0.1; h( q12 ) = When q22 = r22 =0.3; h( q22 ) =

r21 -{( r11 ∧ r21 )+( r31 ∧ r21 )}/2 = 0.0 r22 -{( r12 ∧ r22 )+( r32 ∧ r22 )}/2 = 0.0

When q32 = r23 =0.8; h( q32 ) = r23 -{( r13 ∧ r23 )+( r33 ∧ r23 )}/2 = 0.55 Max{ h( q12 ) , h( q22 ) , h( q32 ) }= h( q32 ) ; Return q32 = 0.8 and k=3 Evaluate the largest qij for all i except k = 3

q12 = Min( r13 , r23 , r33 ) = 0.2; q22 = Min( r13 , r23 , r33 ) = 0.2; j =3 Evaluate_ qkj (j) When q13 = r31 =0.1; h( q13 ) = When q23 = r32 =1.0; h( q23 ) =

r31 -{( r11 ∧ r31 )+( r21 ∧ r31 )}/2 = 0.0 r32 -{( r12 ∧ r32 )+( r22 ∧ r32 )}/2 = 0.65

When q33 = r33 =0.2; h( q33 ) = r33 -{( r13 ∧ r33 )+( r23 ∧ r33 )}/2 = 0.0 Max{ h( q13 ) , h( q23 ) , h( q33 ) }= h( q23 ) ; Return q23 =1.0 and k=2 Evaluate the largest qij for all i except k = 2

q13 = Min( r12 , r22 , r32 ) = 0.3; q33 = Min( r12 , r22 , r32 ) = 0.3;

An Efficient Algorithm to Computing Max-Min Post-inverse Fuzzy Relation

3.2

517

Example

r r  11 12 Given R = r21 r22 r  31 r32 Algorithm in Table-1.

r13  0.9 0.4 0.3  r23  =  0.1 0.3 0.8 , we now provide a trace of the r33   0.1 1.0 0.2

0.9 0.2 0.3 ∴ Q =  0.1 0.2 1.0  .  0.1 0.8 0.3

References 1. Arnould, T., Tano, S.: Interval-valued fuzzy backward reasoning. IEEE Trans. Fuzzy Systems 3(4), 425–437 (1995) 2. Arnould, T., Tano, S.: The inverse problem of the aggregation of fuzzy sets. Int. J. Uncertainity Fuzzyness Knowledge Based System 2(4), 391–415 (1994) 3. Arnould, T., Tano, S., Kato, Y., Miyoshi, T.: Backward chaining with fuzzy “if..then...rules”. In: Proc. of the Second IEEE International Conference on Fuzzy Systems, San Francisco, CA, pp. 548–533 (1993) 4. Bender, E.A.: Mathematical Methods in Artificial Intelligence. IEEE Computer Society Press Silver Spring, MD (1996) 5. Bourke, M.M., Fisher, D.G.: Solution algorithms for fuzzy relational equations with maxproduct composition. Fuzzy Sets and Systems 94, 61–69 (1998) 6. Cen, J.: Fuzzy matrix partial ordering and generalized inverses. Fuzzy Sets and Systems 105, 453–458 (1999) 7. Cheng-Zhong, L.: Generalized inverses of fuzzy matrix. Approximate Reasoning in Decis. Anal., 57–60 (1982) 8. Czogala, E., Drewniak, J., Pedrycz, W.: Fuzzy relation equations on a finite set. Fuzzy Sets and Systems 12, 89–101 (1982) 9. Di Nola, A., Sessa, S.: On the set of solutions of composite fuzzy relation equations. Fuzzy Sets and Systems 9, 275–285 (1983) 10. Drewniak, J.: Fuzzy relation inequalities. Cybernet. Syst. 88, 677–684 (1988) 11. Drewniak, J.: Fuzzy relation equations and inequalities. Fuzzy Sets and Systems 14, 237– 247 (1984) 12. Gottwald, S.: Generalized solvability criteria for fuzzy equations. Fuzzy Sets and Systems 17, 285–296 (1985) 13. Gottwald, S., Pedrycz, W.: Solvability of fuzzy relational equations and manipulation of fuzzy data. Fuzzy Sets and Systems 18, 45–65 (1986) 14. Higashi, M., Klir, G.J.: Resolution of finite fuzzy relation equations. Fuzzy Sets and Systems 13, 65–82 (1984) 15. Kim, K.H., Roush, F.W.: Generalized fuzzy matrices. Fuzzy Sets and Systems 4, 293–315 (1980) 16. Klir, G.J., Folger, T.A.: Fuzzy Sets, Uncertainity, and Information, ch. 1. Prentice-Hall, Englewood Cliffs (1988)

518

S. Chakraborty, A. Konar, and R. Janarthanan

17. Konar, A.: Computational Intelligence: Principles, Techniques and Applications. Springer, Heidelberg (2005) 18. Kosko, B.: Fuzzy Engineering. Prentice-Hall, Englewood Cliffs (1997) 19. Leotamonphong, J., Fang, S.-C.: An efficient solution procedure for fuzzy relational equations with max-product composition. IEEE Trans. Fuzzy Systems 7(4), 441–445 (1999) 20. Lettieri, A., Liguori, F.: Characterization of some fuzzy relation equations provided with one solution in a finite set. Fuzzy Sets and Systems 13, 83–94 (1984) 21. Li, X., Ruan, D.: Novel neural algorithms based on fuzzy δ -rules for solving fuzzy relational equations, part III. Fuzzy Sets and Systems 109, 355–362 (2000) 22. Luoh, L., Wang, W.-J., Liaw, Y.-K.: Matrix-pattern-based computer algorithm for solving fuzzy relational equations. IEEE Trans. on Fuzzy Systems 11(1) (February 2003) 23. Miyakoshi, M., Shimbo, M.: Solutions of composite fuzzy relational equations with triangular norms. Fuzzy Sets and Systems 16, 53–63 (1985) 24. Pappis, C.P.: Multi-input multi-output fuzzy systems and the inverse problem. European J. of Operational Research 28, 228–230 (1987) 25. Pappis, C.P., Adamopoulos, G.I.: A computer algorithm for the solution of the inverse problem of fuzzy systems. Fuzzy Sets and Systems 39, 279–290 (1991) 26. Pappis, C.P., Sugeno, M.: Fuzzy relational equation and the inverse problem. Fuzzy Sets and Systems 15(1), 79–90 (1985) 27. Pedrycz, W.: Inverse problem in fuzzy relational equations. Fuzzy Sests and Systems 36, 277–291 (1990) 28. Pedrycz, W.: Fuzzy Set Calculus. In: Ruspini, E.H., Bonissone, P.P., Pedrycz, W. (eds.) Handbook of Fuzzy Computation. IOP Publishing, Bristol (1998) 29. Pedrycz, W.: Fuzzy relational equations with generalized connectives and their applications. Fuzzy Sets and Systems 10, 185–201 (1983) 30. Perfilieva, I., Tonis, A.: Compatibility of systems of fuzzy relation equations. Int. J. General Systems 29(4), 511–528 (2000) 31. Prevot, M.: Algorithm for the solution of fuzzy relations. Fuzzy Sets and Systems 5, 319– 322 (1981) 32. Rich, E., Knight, K.: Artificial Intelligence. McGraw-Hill, New York (1991) 33. Ross, T.J.: Fuzzy Logic with Engg. Applications. McGraw-Hill, New York (1995) 34. Saha, P.: Abductive Reasoning with Inverse fuzzy discrete Relations, Ph. D. Dissertation, Jadavpur University, India (2002) 35. Saha, P., Konar, A.: A heuristic algorithm for computing the max-min inverse fuzzy relation. Int. J. of Approximate Reasoning 30, 131–147 (2002) 36. Sanchez, E.: Resolution of composite fuzzy relation equations. Inf. Control 30, 38–48 (1976) 37. Sanchez, E.: Solution of fuzzy equations with extended operations. Fuzzy Sets and Systems 12, 237–247 (1984) 38. Sessa, S.: Some results in the setting of fuzzy relation equation theory. Fuzzy Sets and Systems 14, 281–297 (1984) 39. Togai, M.: Application of fuzzy inverse relation to synthesis of a fuzzy controller for dynamic systems. In: Proc. 23rd Conf. on Decision and Control, Las Vegs, NV (December 1984) 40. Wang, S., Fang, S.-C., Nuttle, H.L.W.: Solution set of interval valued fuzzy relational equation. Fuzzy Optim. Decision Making 2(1), 41–60 (2003)

An Efficient Algorithm to Computing Max-Min Post-inverse Fuzzy Relation

519

41. Wu, Y.-K., Guu, S.-M.: An efficient procedure for solving a fuzzy relational equation with Max-Archimedean t-norm composition. IEEE Trans. on Fuzzy Systems 16(1) (February 2008) 42. Yeh, C.-T.: On the minimal solutions of max-min fuzzy relational equations. Fuzzy Sets and Systems 159(1) (January 2008) 43. Zadeh, L.A.: Knowledge representation in fuzzy logic. IEEE Trans. Knowledge Data Engg. (1988) 44. Zimmerman, H.J.: Fuzzy Set Theory and its Applications. Kluwer Academic Publishers, Dordrecht (1996) 45. Chakraborty, S., Konar, A., Jain, L.C.: An Efficient Algorithm to Computing Max-Min Inverse Fuzzy Relation for Abductive Reasoning. IEEE Trans. Sys. Man Cybernet (A) 40(1) (January 2010)

Fuzzy-Controlled Energy-Efficient Weight-Based Two Hop Clustering for Multicast Communication in Mobile Ad Hoc Networks Anuradha Banerjee1, Paramartha Dutta2, and Subhankar Ghosh3 1

Kalyani Govt. Engg. College, Kalyani, Nadia West Bengal, India [email protected] 2 Visva-Bharati University, Santiniketan, West Bengal, India [email protected] 3 Regent Education and Research Foundation, West Bengal, India [email protected]

Abstract. A mobile ad hoc network is an infrastructure-less network where the nodes are free to move independently in any direction. The technique of clustering is used to divide a large network into several sub-networks. Inherent dynamism of nodes gives rise to unpredictable topological structure in ad hoc networks which complicates the clustering process. Moreover, since mobile nodes are battery powered, energy saving is required to increase node lifetime and maintain network connectivity. In this paper we propose a fuzzy-controlled energy-efficient weight-based clustering algorithm (FEW-CA) for mobile ad hoc networks. A fuzzy controller named Cluster-head Elector (CE) is embedded in each node ni which computes weight of node ni. If ni is not already member of any cluster and its weight is greater than a pre-defined threshold, then it is elected as a cluster-head and a cluster is formed with 1-hop and 2-hop neighbors of ni who are not already members of any other cluster. FEW-CA does not allow any node to become member of more then one cluster simultaneously. Simulation results firmly establish that our algorithm performs better than existing state-of-the-art broadcast algorithms. Keywords: Ad hoc network, Cluster, Energy-efficiency, Fuzzy.

1

Introduction

Various clustering algorithms for ad hoc networks appear in the literature. Among them, Highest Degree (HD) [1] algorithm, Lowest ID heuristics (LID) [2], Distributed Clustering Algorithm (DCA) [3], Weight based Clustering Algorithm (WCA) [4] and Stable and Flexible weight based clustering algorithm (ProWBCA) [5] are mentionworthy. In HD algorithm, a node with highest degree among its 1-hop neighbors, become a clusterhead. Degree is the number of neighbors of a node. On the other hand, in LID, a node with least identification number among its neighbors, become a clusterhead. These protocols are 1-hop and completely ignore the energy and mobility B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 520–529, 2011. © Springer-Verlag Berlin Heidelberg 2011

Fuzzy-Controlled Energy-Efficient Weight-Based Two Hop Clustering

521

of nodes. On the other hand, in DCA, each node is assigned a weight ( a real number ≥ 0) depending on which a node may be a clusterhead or cluster member. A node is chosen to be the clusterhead if its weight is higher than weights of its neighbors. The DCA makes an assumption that the network topology does not change during execution of the algorithm. This is not suitable for highly mobile networks. The weight-based clustering algorithm (WCA) obtains 1-hop clusters with one clusterhead. The election of clusterhead is based on the weight of each node. The election of the clusterhead is based on weight of each node. The algorithm performs with four admissible factors for the clusterhead election and maintenance. The factors are degree difference, summation of distances, mobility and cumulative time. Although WCA has shown better performance tan all the previous algorithms, it also has some drawbacks. Weights of all nodes need to be computed before the clustering process begins. This is not suitable for a dynamic network like ad hoc networks. ProWBCA is a 2-hop weight based clustering algorithm performs well for ad hoc networks. Weights of nodes are based on their number of 1-hop and 2-hop neighbors. However, this protocol is not power-aware or mobility-aware, which are very important from the perspective of network throughput and longevity. In this article, we propose a fuzzy-controlled weight based clustering algorithm that considers all these aspects and constructs stable 2-hop clusters. Simulation results emphasize the effectiveness of our proposed scheme.

2

Overview of FEW-CA

Nodes in an ad hoc network may belong to various multicast groups and during cluster formation; we try to include as many members as possible of same group, into a cluster. This reduces the cost of inter-cluster communication during multicast operations. If all members of a multicast group belong to the same cluster C, then the multicast source needs to send the multicast message only to head of the cluster C only. As far as node degree is concerned, FEW- CA emphasizes on the fact that if the number of ordinary cluster members (ordinary cluster members are those that are neither cluster-heads nor gateway nodes) abruptly increase in a cluster, network throughput drops and system performance degrades [3]. On the other hand, increase in the number of gateway nodes, improve inter-cluster connectivity [1, 3]. In order to inculcate the flavor of power-awareness in FEW-CA, we have enforced another constraint that a node which doesn’t have sufficient battery power cannot compete in the election of cluster-head. Also, the more power-equipped a node is, stronger will be its claim for being a cluster-head. In FEW-CA whenever a node joins a cluster, a member-join message is broadcasted by the respective cluster-head within its cluster. Similarly, when a node leaves a cluster, cluster-head broadcasts a member-lost message. If only stable clusters are constructed then the links from cluster-head to its 1-hop neighbors and links from 1-hop neighbors of the cluster-head to 2-hop neighbors of the clusterhead wont break frequently and the cost of member-join and member-lost messages will be significantly reduced.

522

A. Banerjee, P. Dutta, and S. Ghosh

All these observations are in the form of if-then rules which are basic unit of fuzzy function approximation. Advantages of fuzzy logic are that it is flexible, conceptually easy to understand and based on natural language. Moreover, it is tolerant of imprecise data and can model non-linear functions of arbitrary complexity. All these encouraged us to design the scheme of FEW-CA using fuzzy logic. In FEW-CA, each node periodically broadcasts HELLO message within its radiorange and its 1-hop neighbors reply with ACK or acknowledgement message. Attributes of these HELLO message generated by node ni are as follows: i) node identification number ni ii) radio-range Ri iii) geographical position (xi(t), yi(t)) of ni at time t in terms of latitude and longitude iv) velocity vi(t) of ni at time t v) timestamp t vi) cluster-head status (set to 1 if ni is a cluster-head, otherwise it is set to 0) vii) multicast group g of which ni is a member (g = 0 if ni does not belong to any multicast group) After receiving the HELLO message of ni, its neighbours reply with the ACK (acknowledgement) message. The attributes of ACK message transmitted by a neighbour nj of ni at time t consists of the following information: i) source identification number nj ii) destination identification number ni iii) velocity vj(t) of nj at time t iv) geographical position (xj(t), yj(t)) of nj at time t in terms of latitude and longitude v) identification number of heads of those clusters of which nj is a member vi) multicast group g of which nj is a member (g = 0 if nj does not belong to any multicast group) In case of change of cluster-head a head_change message is flooded within the respective cluster. The attributes of head_change are as follows: i) identification number of the new cluster-head h′ ii) timestamp t Attributes of member-join message are, i) identification number of the new cluster member nk ii) timestamp t of joining the cluster iii) status field (set to 1) Attributes of member-leave message are, i) identification number of the lost cluster member nk ii) timestamp t of leaving the cluster iii) status field (set to 0)

Fuzzy-Controlled Energy-Efficient Weight-Based Two Hop Clustering

3

523

Input Parameters of CE

Below appear the descriptions and mathematical expressions of input parameters of CE. •

Residual Energy Ratio

Residual energy ratio αi(t) of node ni at time t is given by, αi(t) = 1 - ei(t) / Ei

(1)

ei(t) and Ei indicate the consumed battery power at time t and maximum or initial battery capacity of ni, respectively. It may be noted from the formulation in 1) that 0≤ αi(t) ≤1. Values close to 1 enhance capability of ni as a cluster-head. •

Node Degree Influence

The formulation of node degree influence is based on the following concepts i)

ii)

iii)

If an entire multicast group is totally contained within a cluster then it is good for network throughput and lifetime. In that case, if a source node needs to send some message to the group, then it will be sufficient to send the message to head of the cluster encapsulating that group. So, a cluster tries to include as many members as possible of same multicast groups within its territory. If most of the cluster-members and gateway nodes are at 1-hop distance, instead of 2-hop distance from the cluster-head then they will receive messages from the cluster-head at lesser delay. Therefore, it is better for a cluster if most of its members and gateways are at 1-hop distance from the cluster-head. Excessive density of nodes within a cluster greatly increases the cost of intra-cluster communication.

The formulation of node degree influence βi(t) of ni at time t is given by, βi(t) = [ (w_h hp_1i(t) + hp_2i(t)) × ovmi(t)]1/2

(2)

hp_1i(t) = { (1/Di(t)) Σ ρ1i,g(t)} g∈MG

(3)

hp_2i(t) = { (1/Di(t)) Σ ρ2i,g(t)} g∈MG

(4)

If a cluster member is at 1-hop distance from the clusterhead then minimum and maximum possible distance of the node from the clusterhead are 1 and Rmax resulting into the average distance of (1+ Rmax)/2; similarly, as far as 2-hop distance of a cluster member from the clusterhead are concerned, the minimum and maximum values are (1+ Rmin) and 2 Rmax, resulting into the average value {(1+ Rmin)/2 + Rmax}. It is quite

524

A. Banerjee, P. Dutta, and S. Ghosh

practical that if a cluster member is at 1-hop delay from the clusterhead then it will receive messages from clusterhead much faster than 2-hop distant cluster members. w_h is a measure of this advantage of 1-hop neighbors of clusterhead compared to 2hop neighbors of the same, in terms of delay. It is mathematically expressed as, w_h = {(1+ Rmin + 2Rmax} / (1+ Rmax)

(5)

MG is the set of various multicast groups in the network. ρ1i,g(t) indicates the number of nodes belonging to the multicast group g that reside within 1-hop neighborhood of ni at time t. Similarly, ρ2i,g(t) indicates the number of nodes belonging to the multicast group g that are 2-hops away from ni at time t. Di(t) is the number of all nodes within 2-hop neighborhood of ni at time t. Let N(t) be total number of nodes in the network at time t and A be the total network area. Then network density ψ(t) at time t is, ψ(t) = N(t) / A

(6)

Under uniform node density, total number of nodes φi(t) within 2-hop neighborhood of ni at time t is, φi(t) = ψ(t) × π (Ri + Rj)2

(7)

Where Ri is the radio-range of ni and nj is the downlink neighbor of ni having highest radio-range among all the downlink neighbors of ni at time t. Then neighborhood overload immunity ovmi(t) on ni at time t is given by, Di(t) / φi(t) ovmi(t) =

if Di(t) ≤ φi(t) (8)

φi(t) / (Z + Di(t))

otherwise

where Z is the total number of nodes in the network. The expression in (8) is based on the fact that the cluster of ni is not overloaded till Di(t) < φi(t). So, ovmi(t) increases as Di(t) approaches φi(t) and acquires maximum value 1 when Di(t) = φi(t). On the other hand, if Di(t) > φi(t), overload immunity takes a very low fractional value which decreases with increase in Di(t). •

Neighbor Cluster Connectivity

Neighbor cluster connectivity ζi(t) of the cluster with head ni at time t is formulated as, ζi(t) = (gi(t) / Di(t)) exp (1/(nci(t) +1))

(9)

gi(t) is the number of gateway nodes of the cluster with head ni at time t and nci(t) is the number of neighbor clusters of the cluster with head ni at time t. The expression in (9) is based on the concept that it is good to have connectivity with various neighbor clusters than having multiple gateway nodes providing connectivity with same cluster.

Fuzzy-Controlled Energy-Efficient Weight-Based Two Hop Clustering

•

525

Cluster Stability

A 2-hop cluster whose head is ni, will be termed as stable provided its links do not break frequently. Necessary conditions for this are i) velocity of the cluster-head ni relative to its one hop neighbors, is small ii) distance of the 1-hop neighbors of ni from ni is small compared to the radio-range of ni i.e. Ri iii) for each 1-hop neighbor nj of ni, relative velocity of nj with its 1-hop neighbors, is small, and iii) for each 1-hop neighbor nj of ni, distance of nj from its 1-hop neighbors, is small compared to the radio-range of nj i.e. Rj. Among the above mentioned conditions, the first two are more important because if a cluster-head has a stable 1-hop neighborhood and unstable 2-hop neighborhood then chances are high that the cluster will not survive as a two-hop cluster but will survive as a one hop cluster. But, on the other hand, if 1-hop neighborhood of a cluster is unstable then the cluster won’t survive irrespective of the stability of its 2-hop neighborhood. So, in the expression of cluster stability, stability of links from clusterhead to its 1-hop neighbors will dominate compared to stability of the links from 1hop neighbors of cluster-head to 2-hop neighbors of the cluster-head. Hence cluster stability csi(t) of a cluster with head ni at time t is given by, csi(t) = { ∏ wij(t) (  wjk(t)) / (|Bj(t)|+1)} exp (-2 |Bi(t)|) nj∈ Bi(t) nk∈ Bj(t) where wij(t) = 1 - (1-1/(|vi(t) – vj(t)|+1)) exp (distij(t)/Ri)

(10) (11)

For any node ni, vi(t) specifies its velocity at time t; Bi(t) is the set of 1-hop neighbors of ni and distij(t) is the distance between ni and nj at time t. The constant 1 is added in the denominator in expression (10) to avoid zero value in the denominator in the situation when 1-hop neighborhood of a 1-hop neighbor of cluster-head is empty. Similarly, 1 is added in expression (11) to avoid zero value in the denominator when (vi(t) – vj(t)). It is evident from (10) that for any node ni, csi(t) ranges between 0 and 1. Stability of a cluster with head ni increases as csi(t) approaches 1.

4

Rule Bases of CE

The division of input parameters of CE into crisp ranges and the corresponding fuzzy variables are shown in table 1. Table 1. Division of parameters of CE into crisp ranges and corresponding fuzzy variables

Range division of residual energy ratio 0-0.40 0.40-0.60 0.60-0.80 0.80-1.00

Range division of other parameters of CE 0-0.25 0.25-0.5 0.5-0.75 0.75-1.00

Fuzzy variables a1 a2 a3 a4

526

A. Banerjee, P. Dutta, and S. Ghosh

According to the study of discharge curve of batteries heavily used in ad hoc networks, at least 40% (fuzzy variable a1 represents the range 0-0.40) of total charge is required to remain in operable condition; 40%-60% (fuzzy variable a2) of the same is satisfactory, 60%-80% (fuzzy variable a3) is good and the next higher range (i.e. 80%-100% or fuzzy variable a4) is more then sufficient from the perspective of remaining energy. Range division of residual energy ratio follows this concept. Ranges of all other parameters are uniformly divided between 0 and 1; the range 0-0.25 is denoted as a1, 0.25-0.5 is denoted as a2, 0.5-0.75 as a3 and 0.75-1.0 as a4. It may please be noted that the parameters residual energy ratio and cluster stability are extremely important from the perspective of existence of the cluster while the other parameters focus on efficiency of the cluster. Hence, residual energy ratio α and cluster stability cs dominate other input parameters of CE. Table 2 shows the combination of residual energy ratio and cluster stability. Both are given equal weight. The temporary output t1 of table 2 is combined with node degree influence in table 3 generating second temporary output t2. Table 4 shows the combination of t2 and neighbor cluster connectivity generating output node-weight of CE. In tables 3 and 4, the output of previous table dominates the new output. The reason is that, t1 is the combination of two parameters both of which are extremely important from the point of view of existence of a cluster, and the effect propagates from t1 to t2. Table 2. Fuzzy Combination of α and cs producing output t1 α→ cs↓ a1 a2 a3 a4

a1

a2

a3

a4

a1 a1 a1 a1

a1 a2 a2 a2

a1 a2 a3 a3

a1 a2 a3 a4

Table 3. Fuzzy Combination of t1 and β producing temporary output t2 t1→ β↓ a1 a2 a3 a4

a1

a2

a3

a4

a1 a1 a1 a2

a2 a2 a2 a3

a3 a3 a3 a3

a3 a3 a4 a4

Table 4. Fuzzy Combination of t2 and ζ producing output node-weight of CE t1→ ζ↓ a1 a2 a3 a4

a1

a2

a3

a4

a1 a1 a1 a1

a2 a2 a2 a2

a3 a3 a3 a3

a3 a3 a4 a4

Fuzzy-Controlled Energy-Efficient Weight-Based Two Hop Clustering

5

527

Simulation Results

Simulation environment appears in table 5. We compare the performance of FEWCA with the protocols LID, HD and ProWBCA. The performance metrics are message cost, packet delivery ratio, number of clusters and rate of change of cluster by members. ns-2 [6] simulator has been used for the purpose of simulation. Table 5. Simulation Parameters Parameter Network Area Transmission Range Interval between HELLO messages

consecutive

Number of nodes MAC layer Traffic type Maximum number of retries before an acknowledgement is obtained Packet Size

Bandwidth Mobility model

Simulation Time Node velocity

Value 900 × 900 m2 in first ten runs, 2000 × 1000 m2 in nest ten runs, 900 × 3000 m2 in last ten runs 10 – 50 m in first ten runs, 30 – 100 m in next ten runs, 10 – 100 m in last ten runs 20 seconds for first ten simulation runs, 30 seconds for next ten and 45 seconds for last ten simulation runs 100 – 500 IEEE 802.11g Constant bit rate (128 kbps/second) 4

64 bytes in first ten runs, 128 bytes in next ten runs, 256 bytes in last ten runs (in different simulation runs) 1- 4 Mbps in first ten runs, 2 – 7 Mbps in first ten runs, 1-10 Mbps in last ten runs Random waypoint mobility model in first 10 runs, Random walk mobility model in subsequent 10 runs and Gaussian model in last 10 runs 1000 s for each run 5-25 m/s

M e s s age cos t vs Num be r of node s 16000

Message cost

14000 12000

LID

10000

HD

8000

ProWBCA

6000

FEW-CA

4000 2000 0 100

200 300 400 Num be r of node s

500

Fig. 1. Graphical illustrations of message cost vs number of nodes

A. Banerjee, P. Dutta, and S. Ghosh

Packet delivery ratio vs Num ber of nodes

Packet delivery ratio

1.2 1 LID

0.8

HD

0.6

ProWBCA

0.4

FEW-CA

0.2 0 100

200 300 400 Num ber of nodes

500

Fig. 2. Graphical illustrations of packet delivery ratio vs number of nodes

Number of clusterheads

Num be r of clus te rhe ads vs Num be r of nodes 80 60

LID HD

40

ProWBCA FEW-CA

20 0 100

200 300 400 Num be r of nodes

500

Fig. 3. Graphical illustrations of number of clusterheads vs number of nodes Rate of change of cluster by mem bers vs Num ber of nodes

Rate of change of cluster by members

528

10 8

LID

6

HD

4

ProWBCA

2

FEW-CA

0 100

200 300 400 Num ber of nodes

500

Fig. 4. Graphical illustrations of rate of change of cluster vs number of nodes

Fuzzy-Controlled Energy-Efficient Weight-Based Two Hop Clustering

529

Figure 1 shows the comparison of the clustering schemes w.r.t. message cost. Unlike LID, HD and ProWBCA, FEW-CA considers stability of links and prefers only stable links for communication. Hence, the possibility of link breakage and as a result, the need for injecting link repair messages in the network decrease in FEW-CA. The cost of head_change, member_join and member_leave messages is also much less in our proposed algorithm, because the clusterhead has good connectivity with the members. All these significantly contribute in decreasing message cost in FEW-CA than its competitors. Less message cost generates less signal collision improving the packet delivery ratio as evident from figure 2. Figure 3 compares the number of clusterheads of the mentioned algorithms. Since LID and HD construct only single hop clusters, number of clusterheads in them is much higher than FEW-CA and ProWBCA. Please note here that the number of clusterheads in FEW-CA and ProWBCA don’t differ much as both are 2-hop clustering algorithm. Figure 4 illustrates the rate of change of cluster by members of the algorithms LID, HD, ProWBCA and FEWCA. In FEW-CA, the clusterheads maintain stable connectivity with its members. As a result, possibility of leaving a cluster by a cluster member is much less in FEW-CA.

6

Conclusion

This paper has presented a fuzzy-controlled weight-based clustering algorithm FEWCA for mobile ad hoc networks. Compared to conventional clustering algorithms, this proposed 2-hop clustering algorithm is more stable w.r.t. topology changes. The algorithm is power-aware and significantly improves network throughput at much lesser cost.

References 1. Agarwal, R., Motwani, M.: Survey of Clustering Algorithm for MANETS. International Journal on Computer Sc. and Engineering 1(2), 98–104 (2009) 2. Correa, B., Ospina, L., Hincape, R.C.: Survey of clustering techniques for mobile ad hoc networks, pp. 145–161 (2007) ISSN 0120-6230 3. Yu, J.Y., Chong, P.H.J.: A survey of clustering schemes for mobile ad hoc networks. IEEE Communications Surveys and Tutorials 7(1) (2005) 4. Yang, W.D., Zhang, G.Z.: A weight-based clustering algorithm for mobile ad hoc networks. In: IEEE Proc. ICWMC 2007 (2007) 5. Pandi Selvam, R., et al.: Stable and flexible weight-based clustering algorithm in mobile ad hoc networks. International Journal on Computer Science and Information Technologies 2(2), 824–828 (2011) 6. www.isi.edu/nsnam/ns

Automatic Extractive Text Summarization Based on Fuzzy Logic: A Sentence Oriented Approach M. Esther Hannah1, T.V. Geetha2, and Saswati Mukherjee3 1

Anna University, Chennai, India [email protected] 2 College of Engineering, Anna University, Chennai, India [email protected] 3 College of Engineering, Anna University, Chennai, India [email protected]

Abstract.The work presents a method to perform automatic summarization of the text through sentence scoring. We propose a method which utilizes the facilities of fuzzy inference system for the purpose of scoring. Preprocessing of the text is done since this technique has its own importance enabling us to filter high quality text. A thorough review of the concepts of summarization enabled us to make use of a group of features which are very appropriate for automatic text summarization. Experimental results obtained by the proposed system on DUC 2002 data reveal that it works to the optimality with respect to other existing methods, and hence is a concrete solution to text summarization. Index Terms: Sentence scoring, Feature extraction, Fuzzy inference, Summarization.

1

Introduction

The increasing availability of online information has necessitated intensive research in the area of automatic text summarization within the Natural Language Processing Community. Extensive use of internet is one of the main reasons why automatic text summarization draws substantial interest [9]. It provides a solution to the information overload problem which is continually encountered in the digital era. Research on automatic text summarization has a very long history, which dates back to at least 40 years, to the first system built at IBM in 1958 [13]. Many innovative approaches were explored which includes statistical and information-centric approaches, linguistic approaches and their combinations. Though there are a number of statistical and linguistics approaches to extractive text summarization systems, most of these involve tasks that are tedious to perform, time consuming, may not be accurate and inappropriate for a given task. In this work, we propose a method that uses a unique set of all important features, makes use of sound preprocessing of text and provides better precision in sentence scoring, thereby obtaining better and more appropriate summaries. We propose a robust technique for B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 530–538, 2011. © Springer-Verlag Berlin Heidelberg 2011

Automatic Extractive Text Summarization Based on Fuzzy Logic

531

extractive text summarization that optimizes feature extraction by making use of a balanced set of features. The set of features so selected not only depends on a given sentence but is based on the overall centrality of the sentence with respect to the entire document. Although genetic algorithms [26, 27] and neural networks [28] can perform just as well as fuzzy logic in applications relating to learning and retrieving natural language, fuzzy logic [29] has the advantage that it is built atop the structures of qualitative description used in everyday language. Since the basis for fuzzy logic is the basis for human communication, it makes it easier to automate tasks that can be successfully performed by humans [30]. In this paper we provide a fuzzy based approach to extractive text summarization where fuzzy rules are used to generate the score for every sentence based on its sentence features. The scoring enabled us to classify every sentence in the text into one of the three categories namely important, unimportant and average based on the importance of the sentence. The rest of the paper is organized as follows: Section II brings out the existing work done on text summarizations focusing on the contributions that builds the research in this subfield of NLP. Section III progresses to discuss the proposed work under various subsections namely, preprocessing, feature extraction and fuzzy inference system. Section IV discusses the evaluation method we have used and the results are provided. Section V concludes the paper with providing scope for future work.

2

Related Works

With the use of machine learning techniques with NLP, a number of approaches that employed statistical techniques to produce document extracts came into picture. Various researchers used the feature extraction process and extracted features such as ‘frequency of term’ [1], sentence position [31], etc., and some used a linear combination of features such as cue words, keywords, title or heading and sentence location [2][7]. Kupiec et al, used human generated abstracts as training corpus, from which he produced extracts. The feature set included sentence length, fixed phrases, sentence position in paragraph, thematic words and uppercase words [7]. Feature extraction techniques were used to locate the important sentences in the text. Conroy & O’leary in 2001 modeled the problem of extracting a sentence from a document using Hidden Markov Model (HMM) [20]. In 2004, Khosrow Kaikhah et al proposed a new technique for summarizing news articles using a neural network that is trained to learn characteristics of sentences that should be included in the summary of the article [21]. S.P. Yong et al worked on developing an automatic text summarization system combining both statistical approach and neural network to summarize documents [22]. Hsun-Hui Huang, Yau-Hwang Kuo, Horng-Chang Yang at al in 2006, proposed to extract key sentences of a document as its summary by estimating the relevance of sentences through the use of fuzzy-rough sets [23]. Ladda Suanmali et al [19] in 2009 developed a system that generates extractive summaries based on a set of features that represent the sentences in a text. The Fuzzy approach was applied to enable efficient

532

M. Esther Hannah, T.V. Geetha, and S. Mukherjee

categorization of sentences that form the summary. Our system makes use of a unique set of features which we feel is important in weighing the importance of a sentence. This set is obtained by combining features such as numerical data & pronoun feature with the ones used in the above method of text summarization. We build a summarization model that exploits the benefits of fuzzy inference.

3

Proposed Works

The proposed method delivers a technique to perform automatic summarization of the text through sentence scoring based on fuzzy inference system. First, a text, as an input, is preprocessed and high quality text is obtained. Several characteristics of each sentence such as title feature, sentence length, term weight, etc. are taken into consideration and the corresponding feature values are obtained. These feature values are used in the fuzzy inference system to generate score for every sentence based on which overall ranking of each of the sentences is done. The importance of a sentence and its presence or absence in the generated summary is based on these features as a whole, rather than on any specific feature. We have used a well-defined set of fuzzy rules in the fuzzy engine such that the rules are triggered based on the cumulative weight of all the features for every sentence. Thus the proposed method uses a sentence-oriented approach, rather than a feature-oriented one. A set of fuzzy rules are defined, which forms the base for sentence scoring. Figure 1 shows the proposed system architecture. The proposed work is divided into modules, namely preprocessing, feature extraction and sentence scoring using fuzzy inference, which are discussed in the following subsections: Preprocessing

Feature Extraction

Arbitrary Text Fuzzy Interface System

Sentence Scoring

Ranking

Summary Fig. 1. System Architecture

Automatic Extractive Text Summarization Based on Fuzzy Logic 3.1

533

Preprocessing

Preprocessing is done as a means of cleaning the document by removing words that do not contain any information that uniquely identifies a sentence. This module filters the noise away. Preprocessing includes Sentence segmentation, Tokenization, Stop word removal and Word stemming respectively. 3.2

Feature Extraction

For any task of text mining, features play an important role. Features are attributes that attempt to represent data used for the task. We focus on seven features for each sentence, viz. sentence length, title resemblance, thematic words, numerical data, proper noun, term frequency and sentence to sentence similarity. The rest of this section discusses these features and methods of extraction. a) Title feature The number of title words in the sentence contributes to title feature. Titles contain group of words that give important clues about the subjects contained in the document. Therefore if a sentence has higher intersection with the title words, the sentence is more important than others. b) Sentence length The number of words in sentence gives good idea about the importance of the sentence. This feature is very useful to filter out short sentences such as datelines and author names commonly found in articles. The short sentences are not expected to belong to the summary. c) Term weight The term weight feature score is obtained by calculating the average of the TF-ISF (Term frequency, Inverse sentence frequency). The frequency of term occurrences within a document has often been used for calculating the importance of sentence. Inverse term frequency helps to identify important sentences that represent the document. d) Sentence to sentence similarity Similarity between sentences is calculated as follows: for a sentence s, the similarity between s and all other sentences is computed by the cosine similarity measure. The score of this feature for a sentence is obtained by computing the ratio of the summation of sentence similarity of a sentence s with each of the other sentences over the maximum value sentence similarity. e) Proper noun The proper noun feature gives the score based on the number of proper nouns present in a sentence, or presence of named entity in the sentence. Usually sentences that contain proper nouns are considered to be important and these should be included in the document summary.

534

M. Esther Hannah, T.V. Geetha, and S. Mukherjee

f) Thematic word The number of thematic word in sentence is an important feature because terms that occur frequently in a document are probably related to topic. Thematic words are words that capture main topics discussed in a given document. We used the top 10 most frequent content word for consideration as thematic. g) Numerical data This feature gives a score for the number of numerical data in a sentence. The contribution made by this feature to the weight given to a sentence is significant since a sentence that contains numerical data essentially contains important information. Each of these seven features is given a value between ‘0’ and ‘1’. We have used 30 percent compression ratio for the generated summary. 3.3

Sentence Scoring Using Fuzzy Inference

The feature values extracted from the features discussed in the previous section are given to the fuzzy inference system to identify the most important sentences of the given document and these sentences are scored accordingly. 3.3.1 Fuzzy Inference System A fuzzy inference system is a control system based on fuzzy logic which is a mathematical system that analyzes analog input values in terms of logical variables that take on continuous values between 0 and 1 in contrast to classical or digital logic, which operates on discrete values of either 0 or 1 (true or false) [15]. There are various membership function types used in fuzzy logic such as: sigmoid, Gaussian, trapezoidal (trapmf), triangular (trimf), etc. We have used the trapezoidal membership function for fuzzification since it has been used in various systems that make use of fuzzy model [25]. It is also simple and widely used and hence we have used it for the proposed fuzzy inference system. Input given to the fuzzy inference system is fuzzified, and the fuzzified value is later defuzzified to obtain an output that aids to know the importance of every sentence.First, the features extracted are used as input to the fuzzy inference system. The score of the seven features lies between 0 and 1, and they are manually categorized as very low(vl), low(l), average(a), high(h), very high(vh) respectively. Thus if the score of sentence length feature is 0.8, then we classify that feature as very high. These input variables are fuzzified into three fuzzy sets namely less, average and more using the trapezoidal membership function. Fuzzy inference engine is constructed making use of a set of fuzzy rules and this enables us to categorize the sentence into one of the output variables. The ‘if’ part of a rule is called the rule antecedent, and is a description of a process state in terms of a logical combination of atomic fuzzy propositions. The ‘then’ part of the rule is called the rule consequent and is a description of the control output in terms of logical combinations of fuzzy propositions. A sample of the IF-THEN rules is given in equation (8).

Automatic Extractive Text Summarization Based on Fuzzy Logic

IF (vl is less ) AND (l is less ) AND (avg is less) AND (h is more) AND ( vh is more) THEN (sentence is important)

535

(8)

The reverse of fuzzification is called defuzzification. Based on the importance of a sentence, the fuzzy system defuzzifies every sentence into one of the three outputs variables namely unimportant, average, and important. This helps us to identify whether a sentence should be present in the summary or not. Such a categorization aids us in summarizing a text document. Once the sentences are categorized, we are able to generate an information-rich summary by taking into account only sentences that are ranked as ‘important’. The top ‘important’ sentences chronologically arranged are used to form the summary, based on the compression ratio that is required. In cases where the needed summary size is not achieved, ‘average’ important sentences are also included in the summary. However “unimportant” are not considered even in such cases.

4

Evaluation and Results

The TIPSTER program with its two main evaluation style conference series TREC & Document Understanding Conference-DUC (now called as Text Analysis Conference-TAC) have shaped the scientific community in terms of performance, research paradigm and approaches. We used 55 documents from DUC2002 for testing the system. Model summaries provided by DUC2002 were used to evaluate our system. Precision, recall, and F-measure are used in the evaluation which can be calculated as follows using equations (9): 5HFDOO 6UHIŀ6V\V 3UHFLVLRQ 6UHIŀ6V\V )VFRUH Į 5HFDOO 3UHFLVLRQ 6V\V6UHI5HFDOOĮ 3UHFLVLRQ

(9)

where Sref and Ssys denote the number of segments appeared in the reference summary and in the system generated summary, respectively. For F-measure, the experiment uses F1 (i.e., the value of α is 1). A set of metrics called ‘Recall Oriented Understudy for Gisting Evaluation (ROUGE)’ [12], which has become the standards of automatic evaluation of summaries. ROUGE calculation is based on various statistical metrics by counting overlapping units such as n-grams, word sequences, and word pairs between systems which generate summaries correlating with those extracted by human evaluations. ROUGEN is an N-gram recall between an automatic summary and a set of manual summaries. Among the different values of N, unigram-based ROUGE score (ROUGE-1) has been shown to agree with human judgments the most [24]. The Precision, Recall and F-score values of the compressed text are generated using ROUGE-1, which is claimed to suit extractive summarization systems better. We have the results of sentence oriented summarizer that uses fuzzy approach and the results are promising. Table 1 shows the results of the proposed system.

536

M. Esther Hannah, T.V. Geetha, and S. Mukherjee

Table 1. Recall, Precision and F-Score Table 2. R, P and F values of proposed system values of sentence oriented system with other methods 6HQWHQFH2ULHQWHG $SSURDFK 1RRIGRFXPHQWV

)6FRUH

6XPPDUL]HU 9DOXH 5HFDOO 3UHFLVLRQ )VFRUH                    !    

%DVHOLQH 6HQWHQFH )HDWXUH RI 528*( 2ULHQWHG RULHQWHG '8& $SSURDFK $SSURDFK 

5HFDOO

06 ZRUG 

   

3UHFLVLRQ     )VFRUH    

The Fuzzy based sentence oriented summarizer system produced an average Recall of 0.4918, an average Precision of 0.4734 and an average F-score of 0.4824. Table-2 compares the average Recall, Precision and F-score results for ROUGE-1 produced by our sentence-oriented approach with the following: (i) Feature oriented approach using fuzzy (ii) Baseline of DUC 2002 (iii) MS-Word 2007.

0 .4 9

0.4 7

Sent ence Orient ed A p proach

0.4 5

Feat ure orient ed A p proach B aseline o f DUC20 02

0 .4 3

M icro so f t wo rd 20 07

0.4 1

0.49 0.48 0.47 0.46 0.45 0.44 0.43 0.42 0.41 0.4 0.39

Sentence Oriented Approach Feature oriented Approach Baseline of DUC2002 Microsoft w ord 2007 F-score

0 .3 9 Recall

Fig. 2. Comparison of Recall Sentence Oriented Approach

0.49 0.47

Feature oriented Approach

0.45

Baseline of DUC2002

0.43

Fig. 3. Comparison of F-score 0.5 Sentence Oriented Approach

0.48 0.46

Feature oriented Approach

0.44 Baseline of DUC2002

0.42

0.41 0.4

0.39 Precision

Microsoft w ord 2007

Fig. 4. Comparison of Precision

Microsoft word

0.38 Recall

Precision F-score

Fig. 5. Comparison of Recall, Precision, F-score value

Automatic Extractive Text Summarization Based on Fuzzy Logic

537

The above results clearly show that the proposed system promises to be better than the existing ones for extractive text summarization. The Graphical representation of the results is as follows:

5

Conclusion

The proposed method provides a fuzzy based approach to text summarization, where the system selects the right set of candidates for the summary. It also can improve the quality of summarization since it extracts sentences using the scores derived from fuzzy inference system. Experimental results show that the proposed method outperforms other automatic text summarization methods. Testing this scoring methodology for other subtasks of text mining is a possible future work. We believe that by using our approach for sentence scoring and by building solutions for summarization that adhere to our proposal, redundant work that typically characterizes summarization can be avoided.

References 1. Luhn, H.P.: The Automatic Creation Of Literature Abstract. IBM Journal of Research and Development 2, 159–165 (1958) 2. Edmundson, H.P.: New Methods in Automatic extracting. Journal of the Association for Computing Machinery 16(2), 264–285 (1969) 3. Salton, G., Buckley, C.: Term_Weighting Approaches in Automatic Text Retrieval. Information Processing and Management 24, 513–523 (1997) 4. Salton, G.: Automatic Text Processing: The, Analysis, Transformation, and Retrieval of Information by Computer. Addison-Wesley Publishing Company (1989) 5. Fattah, M.A., Ren, F.: Automatic Text Summarization. Proceedings of World Academy of Science, Engineering and Technology 27, 192–195 (2008) 6. Lin, C.Y.: Training a Selection Function for Extraction. In: Proceedings of the Eighth International Conference on Information and Knowledge Management, Kansas City, Missouri, United States, pp. 55–62 (1999) 7. Kupiec, J., Pedersen, J., Chen, F.: A Trainable Document Summarizer. In: Proceedings of the Eighteenth Annual International ACM Conference on Research and Development in Information Retrieval (SIGIR), Seattle, WA, pp. 68–73 (1995) 8. Kulkarni, A.D., Cavanaugh, C.D.: Fuzzy Neural Network Models for Classification. Applied Intelligence 12, 207–215 (2000) 9. Mani, I., Maybury, M.: Advances in automatic text summarization. MIT Press (1999) 10. Radev, D.R., Hovy, E., McKeown, K.: Introduction to the special issue on summarization, vol. 28(4), pp. 399–408 (2002) 11. Mani, I., Klein, G., House, D., Hirsctman, L., Firmin, T., Sundheim, B.: SUMMAC: a text summarization evaluation. Natural Language Engineering 8(1), 43–68 (2002) 12. Lin, C.-Y.: Rouge: A package for automatic evaluation of summaries. In: Proceedings of the ACL 2004 Workshop, pp. 74–81 (2004) 13. Fischer, S., Roark: Query-focused summarization by supervised sentences ranking and skewed word distribution. In: Proceedings of DUC (2006)

538

M. Esther Hannah, T.V. Geetha, and S. Mukherjee

14. Freund, Y., Scaphire, R.E.: Experiments with a new boosting algorithm. In: Proceedings of the Thirteenth International Conference on Machine Learning (1996) 15. Pal, S.K., Mithra, P.: Case generation using rough sets with fuzzy representations. IEEE Transactions on Knowledge and Data Engineering 16(3) (March 2004) 16. Dubois, D., Prade, H.: Putting rough sets and fuzzy sets together. Intelligent Decision Support, 203–232 (1992) 17. Svore, K., Vanderwende, L., Burges, C.: Enhancing single document summarization by combining RankNet and third-party sources. In: Proceedings of the Joint Conference on Empirical Methods in Natural Language Processing and Computational Natural Language Learning, pp. 448–457. Association for Computational Linguistics, Prague (2007) 18. Radev, D.R., Hovy, E., McKeown, K.: Introduction to the special issue on summarization. Computational Linguistics 28(4), 399–408 (2002) 19. Suanmali, L., Binwahlan, M.S., Salim, N.: Sentence Features Fusion for Text Summarization using Fuzzy Logic. In: Proceedings of Ninth International Conference for Text Summarization Using Fuzzy Logic, pp. 142–146 (2009) 20. Conroy, J.M., O’leary, D.P.: Text summarization via hidden markov Models. In: Proceedings of SIGIR 2001, pp. 406–407 (2001) 21. Kaikhah, K.: Text Summarization Using Neural Networks. Faculty Publications, Texas State University (2004) 22. Yong, S.P., Abidin, A.I.Z., Chen, Y.Y.: A Neural Based Text Summarization System. In: Proceedings of the 6th International Conference of Data Mining (2005) 23. Huang, H.-H., Kuo, Y.-H., Yang, H.-C.: Fuzzy-Rough Set Aided Sentence Extraction Summarization. In: Proceedings of the First International Conference on Innovative Computing, Information and Control (2006) 24. Lin, C.Y., Hovy, E.H.: The Potential and Limitation of Sentence Extraction for Summarization. In: Proceedings of the HLT/NAACL Workshop on Automatic Summarization, Edmonton, Canada (2003) 25. Rubens, N.O.: The Application of Fuzzy Logic To The Construction Of The Ranking Function of Information Retrieval Systems. Computer Modelling and New Technologies 10(1), 20–27 (2006) 26. Sakakibara, Y.: Learning context-free grammars using tabular representations. Pattern Recognition 38(9), 1372–1383 (2005) 27. Keller, B., Lutz, R.: Evolutionary induction of stochastic context free grammars. Pattern Recognition 38(9), 1393–1406 (2005) 28. Sakakibara, Y., Golea, M.: Simple recurrent networks as generalized hidden markov models with distributed representations. In: Proceedings of IEEE International Conference on Neural Networks (ICNN 1995), pp. 979–984. IEEE Computer Society Press, New York (1995) 29. Wang, X., Chaudhari, N.S.: Alignment based Similarity Measure for Grammar Learning. In: Proceedings of IEEE International Conferenceon Fuzzy Systems (Fuzz-IEEE 2006), pp. 9034–9041 (2006) 30. Kyoomarsi, F., Khosravi, H., Eslami, E., Dehkordy, P.K., Tajoddin, A.: Optimizing Text Summarization Based on Fuzzy Logic. In: The Proceedings of the Seventh IEEE/ACIS International Conference on Computer and Information Science. IEEE computer Society (2008) 31. Baxendale, P.: Machine-made index for technical literature - an experiment. IBM Journal of Research Development, 354–361 (1958)

An Improved CART Decision Tree for Datasets with Irrelevant Feature Ali Mirza Mahmood1, Mohammad Imran2, Naganjaneyulu Satuluri1, Mrithyumjaya Rao Kuppa3, and Vemulakonda Rajesh4 1

Acharya Nagarjuna University, Guntur, Andhra Pradesh, India 2 Rayalaseema University, Kurnool, Andhra Pradesh, India 3 Vaagdevi College of Engineering, Warangal, Andhra Pradesh, India 4 Pursing M.Tech, MIST, Sathupalli, Khamaman District, Andhra Pradesh, India [email protected]

Abstract. Data mining tasks results are usually improved by reducing the dimensionality of data. This improvement however is achieved harder in the case that data size is moderate or huge. Although numerous algorithms for accuracy improvement have been proposed, all assume that inducing a compact and highly generalized model is difficult. In order to address above said issue, we introduce Randomized Gini Index (RGI), a novel heuristic function for dimensionality reduction, particularly applicable in large scale databases. Apart from removing irrelevant attributes, our algorithm is capable of minimizing the level of noise in the data to a greater extend which is a very attractive feature for data mining problems. We extensively evaluate its performance through experiments on both artificial and real world datasets. The outcome of the study shows the suitability and viability of our approach for knowledge discovery in moderate and large datasets. Keywords: Classification, Decision trees, Filter, Randomized gini index.

1

Introduction

In Machine Learning community, and in Data Mining works, Classification has its own importance. Classification is an important part and the research application field in the data mining [1]. With ever-growing volumes of operational data, many organizations have started to apply data-mining techniques to mine their data for novel, valuable information that can be used to support their decision making [2]. Organizations make extensive use of data mining techniques in order to define meaningful and predictable relationships between objects [3]. Decision tree learning is one of the most widely used and practical methods for inductive inference [4]. Decision trees are one of the most effective machine learning approach for extracting practical knowledge from real world datasets [5]. The main contributions of this work can be summarized as follows. (i)We show that a fast random sampling framework can be used to enhance the generalization accuracy of the tree. (ii) It is worth to note here that the main peculiarity of this composite splitting criterion is that the resulting decision tree is B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 539–549, 2011. © Springer-Verlag Berlin Heidelberg 2011

540

A.M. Mahmood et al.

better in accuracy. (iii) We connect the theoretical results from state-of-the-art decision tree algorithm (CART) showing the viability of our method and also show empirical results supporting our claim.

2

Related Work

In this Section, we present some recent work on decision trees in different areas, Aviad. B [6] have proposes and evaluates a new technique to define decision tree based on cluster analysis. The results of the model were compared to results obtained by conventional decision trees. It was found that the decision rules obtained by the model are at least as good as those obtained by conventional decision trees. In some cases the model yields better results than decision trees. In addition, a new measure is developed to help fine-tune the clustering model to achieve better and more accurate results. Pei-Chann Chang [7] have applied fuzzy logic as a data mining process to generate decision trees from a stock database containing historical information. They have establishes a novel case based fuzzy decision tree model to identify the most important predicting attributes, and extract a set of fuzzy decision rules that can be used to predict the time series behavior in the future. The fuzzy decision tree generated from the stock database is then converted to fuzzy rules that can be further applied in decision-making of stock price’s movement based on its current condition. Leyli Mohammad Khanli [8] have applied active rule learning is regarded for resource management in grid computing. Rule learning is very important for updating rules in active database system. But, it is also very difficult because of lacking methodology and support. Decision tree can use into rule learning to cope with the problems arisen in active semantic extraction, termination analysis of rules set and rules update. Also their aim from rule learning is learning new attributes in rules such as time, load balancing regarded to instances of real Grid environment that decision tree can provide it. Ali Mirza Mahmood [9] have proposed the use of expert knowledge in pruning decision trees for applicability in medical analysis. There has been significant research interest in decision trees in recent years. In [10] author have proposed an improved decision tree classification algorithm MAdaBoost which constructs cascade structures of more decision tree classifiers based on AdaBoost for tackling the problem of imbalanced datasets. The improved algorithm eliminates the short coming of imbalance datasets and improves the overall accuracy of cascade classifiers. In [11] author proposed improved decision tree which uses series of pruning techniques that can greatly improve construction efficiency of decision trees when using for uncertain data.

3

Components of Randomized Gini Index

In this Section, we investigate to propose a new Randomized Gini Index framework (RGI). Our randomized sampling method depends on small random subset of attributes. We assume that the subset of the training data is small, i.e. it is computationally cheap to act on such a set in a reasonable time. Also, such randomized sampling is done multiple times. We focus on a set of commonly used random sampling procedure and Filter. Next, we try to adapt and deploy them as RGI components. The next stage of RGI tries to consider both gini index and weights for

An Improved CART Decision Tree for Datasets with Irrelevant Feature

541

splitting of attributes. The quality of solution fine-tuning, mainly, depends on the nature of the filter involved and the parameters of random sampling. The following four sub sections, detail different design alternatives for both random sampling and filter procedure search for RGI components. 3.1

Random Sampling Method

Due to the large dimensionality of the feature space, it may be difficult for a search method to search appropriate features. In order to increase the computational speed we used random sampling method [12]. Randomized Sampling (RS) is the process of generating random subset datasets from the original dataset where every feature has equal chance. In random sampling we choose a subset of m features out of the presented n features such that m

80

90

100

Fig. 11. Maximum matched points obtained for identical relative mask size

Proposing a CNN Based Architecture of Mid-level Vision

567

On the other hand, the proposed architecture also ensures that even for similar mask size, the number of matched pairs between face images of different subjects is poor. It ensures reduction of false acceptance and false rejection thus strengthening the decision making capabilities of the two parallel visual pathways. The mask size can therefore be automatically varied by the feedback driven by the recognition score (Yij) with feedback transfer function A, here defined as flexibility parameter, which changes the input mask size B of the first layer of the CNN. The proposed F-CNN is used for maximizing the accuracy of the two pathways by increasing the number of matched points between peer face images (same subject with PIE variation) and rejecting wrong pair of face images (different subjects) successfully as tabulated in Table 1. Here all the face images are normalized to 128×128 pixels, so when the flexile mask size of the input layer of the CNN converges to the mask size of the peer match of PIE varied face image, then maximum number of identical key points are generated indicating enhancement of the feed forward paths in the central visual pathway as shown in Fig. 11.

5

Conclusion

The proposed CNN based architecture model is inspired from the functionality of mid-level vision. Early signatures of patterns can be interpreted by the proposed network. The experimental results are quite in agreement with the fact that the middle level visual system feeds to the WHERE and WHAT visual pathways significantly. In future, the relative size of the pattern of interest present in an image could be taken care of by the feedback to make the proposed CNN further flexible. Also, attempts may be made to model the depth information so that 3D recognition is made possible by the CNN based architecture. The dichotomy between the conscious (WHAT) and unconscious (WHERE) modules in visual processing can also be addressed in future using the proposed network.

References 1. Marr, D.: Vision: A computational investigation into the human representation & processing of visual information. MIT Press (2010) 2. Ungerleider, L.G., Mishkin, M.: Two Cortical Visual Systems. In: Ingle, D.J., Goodale, M.A., Mansfield, R.J.W. (eds.) Analysis of Visual Behavior, pp. 549–586. The MIT Press, Cambridge (1982) 3. Rodieck, R.W., Stone, J.: Analysis of receptive fields of cat retinal ganglion cells. Journal of Neurophysiology 28, 833–849 (1965) 4. Chua, L.O., Roska, T.: Cellular Neural Networks and Visual Computing. Cambridge University Press (2002) 5. Itti, L., Koch, C., Niebur, E.: A model of saliency based visual attention for rapid scene analysis. IEEE Trans. on PAMI 20, 1254–1259 (1998) 6. Koenderink, J.J.: The structure of images. Biological Cybernetics 50, 363–396 (1984) 7. Lindeberg, T.: Scale-space theory: A basic tool for analyzing structures at different scales. Journal of Applied Statistics 21(2), 224–270 (1994)

568

A. Das, A. Roy, and K. Ghosh

8. Livingstone, M.S., Hubel, D.H.: Anatomy and physiology of a colour system in the primate visual cortex. J. Neurosci. 4, 309–356 (1984) 9. Livingstone, M.S., Hubel, D.H.: Segregation of form, colour, movement, and depth: anatomy, physiology, and perception. Science 240, 740–749 (1988) 10. Kandel, E.R., Schwartz, J.H., Jessel, T.M.: Principles of Neural Science, 3rd edn. Elsevier, New York (1991) 11. Lowe, D.G.: Object recognition from local scale-invariant features. Proceedings of the International Conference on Computer Vision 2, 1150–1157 (1999) 12. Lowe, D.G.: Distinctive Image Features from Scale-Invariant Key points. International Journal of Computer Vision 60(2), 91–110 (2004) 13. http://www.klab.caltech.edu/codedata/codedata.shtml

Multithreaded Memetic Algorithm for VLSI Placement Problem Potti Subbaraj1 and Pothiraj Sivakumar2 1

Sri Nandhanam College of Engineering & Technology, Tirupattur, Vellore District, TamilNadu - 635601, India [email protected] 2 Electronics and Communication Engineering Department, Arulmigu Kalasalingam College of Engineering, Krishnankoil, Tamil Nadu - 626126, India [email protected] Abstract. Due to rapid advances in VLSI design technology during the last decade, the complexity and size of circuits have been rapidly increasing, placing a demand on industry for faster and more efficient CAD tools. Physical design is a process of converting the physical description into geometric description. Physical design process is subdivided into four problems: Partitioning, Floor planning, Placement and Routing. Placement phase determines the positions of the cells. Placement constrains are wire-length, area of the die, power minimization and delay. For the area and wire length optimization a modern placer need to handle the large–scale design with millions of object. This thesis work aims to develop an efficient and low time complexity algorithms for placement. This can be achieved by the use of a problem specific genotype encoding, and hybrid, knowledge based techniques, which support the algorithm during the creation of the initial individuals and the optimization process. In this paper a novel memetic algorithm, which is used to solve standard cell placement problem is presented. These techniques are applied to the multithread of the VLSI cell placement problem where the objectives are to reduce power dissipation and wire length while improving performance (delay). Keywords: VLSI design, physical design, placement, standard cell, multithread.

1

Introduction

In the past thirty years, VLSI-CAD (Computer-Aided Design of Very Large-Scale Integrated circuits) has been an enabling force behind the exponential growth of the performance and capacity of integrated circuits [6]. The most common way of breaking up the layout problem into sub-problems is first to do logic partitioning where a large circuit is divided into a collection of smaller modules according to some criteria, then to perform component placement, and finally determine the approximate course of the wires in a global routing phase after which a detailed-routing phase determines the exact course of the wires in the layout area. The main objectives of the placement design process are to minimize the total chip area and the total estimated wire length for all the nets. Optimization of the chip area usage can fit more functionality into a given chip area. Optimization of the total estimated wire length B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 569–576, 2011. © Springer-Verlag Berlin Heidelberg 2011

570

P. Subbaraj and P. Sivakumar

can reduce the capacitive delays associated with longer nets and speed up the operation of the chip. Thus, the placement design process has a pronounced affect on the final chip performance. Since module placement is an NP-hard problem, therefore, it cannot be solved exactly in polynomial time [5, 7]. Trying to get an exact solution by evaluating every possible placement to determine the best one would take time proportional to the factorial of the number of modules. Consequently, it is impossible to use this method for circuits with any reasonable number of modules. To search through a large number of candidate placement configurations efficiently, a heuristic algorithm must be used [2]. The traditional approach in placement is to construct a global placement (initial placement) by using constructive placement heuristic algorithms. A detailed placement follows to improve the initial placement. A modification is usually accepted if a reduction in cost occurs, otherwise it is rejected. Global placement produces a complete placement from a partial or non-existent placement. It takes a negligible amount of computation time compared to detailed (iterative improvement) placement and provides a good starting point for them [20]. Usually, global placement algorithms include random placement, cluster growth, partitioning-based placement [9], numerical optimization, and branch and bound techniques [18]. Since partitioning-based methods and numerical optimization methods do not directly attempt to minimize wire length, the solution obtained by these methods is sub-optimal in terms of wire length. Therefore a detailed placement algorithm must be performed to improve the solution. There are two classes of detailed placement methods: Deterministic and Stochastic heuristics. A deterministic heuristic interchanges randomly selected pairs of modules and only accepts the interchange if it results in a reduction in cost [10].

2

Problem Formulation

A mathematical formulation of an approximation to the standard cell placement problem is now presented. Usually, a circuit is represented by a hypergraph G (V, E), where the vertex set V = {v1, v2, v3, …,vn} represent the nodes of the hypergraph (set of cells to be placed), and E = {e1, e2, e3, …,en} represents the set of edges of the Hypergraph (set of nets connecting the cells). The two dimensional placement regions are represented as an array of legal placement locations. The Hypergraph is transformed into a graph (a Hypergraph with all hyperedge sizes equal to 2) via clique model for each net. Each edge ej is an order pair of vertices with a non-negative weight Wij assigned to it. The placement task seeks to assign all cells of the circuit to legal locations such that cells do not overlap. Each cell i is assigned a location on the XY-plane. The cost of an edge connecting two cells i and j, with locations (xi, yj) and (xi, yj) is computed as the product of the squared l2 norm of the difference vector (xi xj ) ( yi - yj ) and the weight of the connecting edge wij The total cost, denoted Φ(x,y) can then be given as the sum of the cost over all edges; i.e: Φ(x,y) = ∑ 2.1

(1)

Wire Length Estimation

It is computationally expensive to determine the exact total wire length for all the nets at the placement stage and therefore, the total wire length is approximated during

Multithreaded Memetic Algorithm for VLSI Placement Problem

571

placement. To make a good estimate of the wire length, we should consider the way in which routing is actually done by routing tools. Almost all automatic routing tools use Manhattan geometry; that is, only horizontal and vertical lines are used to connect any two points. Further, two layers are used; only horizontal lines are allowed in one layer and only vertical lines in the other [20]. An efficient and commonly used method to estimate the wire length is the semiperimeter method [19]. The wire length in this method is approximated by half the perimeter of the smallest bounding rectangle enclosing all the pins. For Manhattan wiring, this method gives the exact wire length for all two-terminal and three-terminal nets, provided that the routing does not overshoot the bounding rectangle. For nets with more pins and more zigzag connections, the semi-perimeter wire length will generally be less than the actual wire length. Moreover, this method provides the best estimate for the most efficient wiring scheme, the Steiner tree. The error will be larger for minimal spanning trees and still larger for chain connections. In practical circuits, however, two and three terminal nets are most common. Thus, the semi-perimeter wire length is considered to be a good estimate [20]. 2.2

String Encoding

Genetic algorithms are powerful optimization techniques that have been successful in solving many hard combinatorial optimization problems [12]. The power of GA’s comes from the fact that the technique is robust, and can deal successfully with a wide

Fig. 1. String Encoding

572

P. Subbaraj and P. Sivakumar

range of problem areas, including those which are difficult for other methods to solve [3]. In this genetic-based placement algorithm, a string is represented by a set of alleles (the number alleles equal to the number of cells). Each allele indicates the index, the x-coordinates and row number of the cell. Fig.2a illustrates the string encoding of the cell placement given in Fig. 2b. 2.3

Scoring Function

Each individual is evaluated to determine its fitness through a scoring function. The scoring function F is the reciprocal of the total HPWL for all the nets. ∑

(2)

Where, HPWL is the sum of the half perimeter of the smallest bounding rectangle for each net. HPWLi is the estimate wire length of net i and n is the number of nets. In the implementation, cell overlaps are removed and row lengths are adjusted before evaluating the chromosome. One reason of doing so is due to the fact that removing the overlaps after every generation not only gives the algorithm a more accurate picture of the wire length but also gives the algorithm repeated chances to optimize the circuit after it has been perturbed by overlap removal [21]. Therefore the row length control and overlap penalty are not considered in the scoring function. 2.4

Initial Population Construction

For each configuration in the initial population, the x-coordinate and row numbers of cells are determined randomly. This kind of constructor can diversify the initial solutions but also tend to have a slower rate of convergence. Thus, some placement solutions produced by Cluster Seed method are injected into the initial population to increase the convergence rate. 2.5

Crossover and Mutation Operator

The Traditional crossover operator used in GA may produce infeasible solutions for the standard cell placement problem, therefore a crossover operator called Order crossover is considered. Fig. 2a shows a one-point order crossover operator where each pair of parents generates two children with a probability equal to the crossover rate. In this method, a single cut point is chosen at random. The crossover operator first copies the array segment to the left point from one parent to one offspring. Then it fills the remaining part of the offspring by going through the other parent, from the beginning to the end and taking those elements that were left out, in order. The two-point order crossover operator is similar to one-point order crossover operator, except that it has to choose two crossover points randomly. An example of two-point order crossover operator is illustrated in Fig. 2b.

Multithreaded Memetic Algorithm for VLSI Placement Problem

573

Fig. 2. One-Point and two-Point Order Crossover

Following crossover, each offspring is mutated with a probability equal to the mutation rate. The mutation operator mutates an individual by interchanging randomly selected pair of cells without changing the x-coordinate and row number. Replacement follows mutation, where the population for next generation is chosen from the combined set of parents and offspring’s. This process is done by replacing the two worst individuals in the population with the offspring’s only if the latter are better than the worst two individuals. This replacement method is also called “elitism” mechanism. One feature of this mechanism is that it ensures the maximum fitness of the population can never be reduced from one generation to the next. 2.6

Hybrid Simulated Annealing (SA)

In this work we have hybridized the genetic algorithm template with the SA method. The SA method is impregnated within GA, between the crossover and mutation operations, to improve all the solutions obtained after the crossover operation and before subjected to mutation operation. SA is very simple technique for State Space Search Problem. It can start from any state. And it is always move to a neighbor with the min cost (assume minimization problem). It can stop when all neighbors have a higher cost than the current state. 2.7

Multithreading

Multithreading (MT) is a technique that allows one program to do multiple tasks concurrently. The basic concept of multithreaded programming has existed in research and development labs for several decades. The threading models we describe are strictly software models that can be implemented on any general-purpose hardware. Much research is directed toward creating better hardware that would be uniquely suited for threaded programming. The memetic algorithm is modified slightly to make it distributed. A number of instances of the genetic algorithm are spawned and run independently an in parallel for number generations. After a set number of generations the separate instances stop and trade solutions with each other to introduce diversity into their populations and keep them from stagnating at local minima. They then repeat this process for a set number of epochs, which can be

574

P. Subbaraj and P. Sivakumar

specified on the command line as well. After all epochs the best solution is chosen from all the instances of the genetic algorithm. To keep the separate instances from reaching the same local minimum only one crossover function is used per instance.

3

Experimental Results

In this work the proposed algorithm is coded in C++ to get a nearer to optimal solution for the VLSI placement problem and experiments were conducted in an Pentium Quad core 2 processor with 2.6 GHz. The outcome of results obtained for popular benchmark circuits ISPD02 was compared with that of the standard Genetic Algorithm methodology which is also coded in C++ and run on the same machine. The following parameters are fixed by conducting trials on various population sizes, crossover probabilities, mutation probabilities and the satisfactory convergent speed exploring scope of the proposed search techniques. Population size=20, Crossover probability (Pc) =0.6, Mutation probability (Pm) =0.01, Number of Threads=4. For all the bench mark circuit taken in this work, the proposed algorithm is able to outperform the standard Genetic algorithm both in terms of number of iterations required to reach a nearer to optimal solution. Table 1. Comparison between Fast Place and MTMA

Circuit

ibm01 ibm02 ibm03 ibm04 ibm05 ibm06 ibm07 ibm08 ibm09 Ibm10 Ibm11 Ibm12 Ibm13 Ibm14 Ibm15 Ibm16 Ibm17 Ibm18

Nets

Pins

14111 19584 27401 31970 28446 34826 48117 50513 60902 75196 81454 77240 99666 152772 186608 190048 189581 201920

50566 81199 93573 105859 126308 128182 175639 204890 222088 297567 280786 317760 357075 548616 715823 778823 860036 819697

HPWL(x10e6) Fast Place MTMA Capo 1.86 1.81 4.06 4.00 5.11 5.09 6.39 6.35 10.56 10.54 5.50 5.45 9.63 9.61 10.26 10.15 10.56 10.50 19.70 19.64 15.73 15.69 25.83 25.74 18.73 18.64 36.69 36.72 43.85 43.79 49.63 49.60 69.07 69.52 47.46 47.41

Run Time Fast Place MTMA Capo 3m 59s 2m 50s 7m 15s 5m 08s 8m 23s 6m 19s 10m 46s 8m 35s 10m 44s 8m 38s 12m 08s 8m 55s 18m 32s 12m 15s 19m 53s 14m 37s 22m 50s 16m 15s 29m 04s 19m 58s 31m 11s 24m 01s 30m 41s 23m 11s 39m 27s 26m 24s 1h 12m 45m 03s 1h 30m 1h 11s 1h 31m 1h 12s 1h 43m 1h 58s 1h 44m 1h 59s

Multithreaded Memetic Algorithm for VLSI Placement Problem

4

575

Conclusion

In this paper, we have presented multithreaded memetic algorithm for circuit placement and the performance was compared with Fast placement. Due to the ability of the Simulated Annealing method it finds the minimized wire length in fewer generations. We applied these algorithms to ISPD02 benchmark circuits from the results shows in Table 1, it is clear that MTMA outperforms than Fast Placement method. The experimental results show that multithreading has good accelerated performance on larger population sizes. A couple of other important factors can help to get the best performance out of our transformation to multithreading such as avoiding of excessive memory transfer, inherent parallelism and computation dependency. More sophisticated cluster approaches to parallelisation will have to be examined in the future.

References 1. Alpert, C.J., Kahng, A.B.: Netlist Partitioning: A Survey.: Integration. The VLSI Journal, 64–80 (1995) 2. Areibi, S.: Iterative Improvement Heuristics for the Standard Cell Placement: A Comparison. In: 5th World Multi Conference on Systemics, Cybernetics and Informatics, Orlando, Florida, pp. 89–94 (2001) 3. Areibi, S., Moussa, M., Abdullah, H.: A Comparison of Genetic/Memetic Algorithms and Other Heuristic Search Techniques. In: International Conference on Artificial Intelligence, Las Vegas, Nevada, pp. 660–666 (2001) 4. Areibi, S., Thompson, M., Vannelli, A.: A Clustering Utility Based Approach for ASIC Design. In: 14th Annual IEEE International ASIC/SOC Conference, Washington, DC, pp. 248–252 (2001) 5. Blanks, J.P.: Near Optimal Quadratic Based Placement for a Class of IC Layout Problems. IEEE Circuits and Devices 1(6), 31–37 (1985) 6. Chang, H., Cooks, L., Hunt, M.: Surviving the SOC Revolution. Kluwer Academic Publishers, London (1999) 7. Donath, W.E.: Complexity theory and design automation. In: 17th Design Automation Conference, pp. 412–419 (1980) 8. Etawil, H., Areibi, S., Vannelli, T.: Convex Programming based Attractor-Repeller Approach for Global Placement. In: IEEE International Conference on CAD, San Jose, California, pp. 20–24 (1999) 9. Garey, M.R., Johnson, D.S.: Computers and Intractability. Freeman, San Francisco (1979) 10. Goto, S., Kuh, E.: An approach to the two-dimensional placement problem in circuit layout. IEEE Trans., Circuits System, CAS 25(4), 208–214 (1976) 11. Hagen, L., Kahng, A.B.: A New Approach to Effective Circuit Clustering. In: IEEE International Conference on CAD, pp. 422–427 (1992) 12. Holland, J.H.: Adaption in Natural and Artificial Systems. University of Michigan, Press, Ann Arbor (1975) 13. Karger, P.G., Preas, B.T.: Automatic Placement: A Review of Current Techniques. In: Proceedings of The 23rd DAC, Las Vegas, Nevada, pp. 622–629 (1986)

576

P. Subbaraj and P. Sivakumar

14. Karypis, G., Aggarwal, R., Kumar, V., Shekhar, S.: Multilevel Hypergraph Partioning: Application in VLSI Design. In: Proceedings of DAC, Las Vegas, Nevada, pp. 526–529 (1997) 15. Kennings, A.: Cell Placement Using Constructive and Iterative Methods.: PhD thesis, University of Waterloo, Ont. Canada (1997) 16. Kernighan, B.W., Lin, S.: An Efficient Heuristic Procedure for Partitioning Graphs. The Bell System Technical Journal 49(2), 291–307 (1970) 17. Mallela, S., Grover, L.K.: Clustering Based Simulated Annealing for Standard Cell Placement. In: Proceedings of The 23rd DAC, Las Vegas, Nevada, pp. 312–317 (1989) 18. Schuler, D.M., Ulrich, E.: Clustering and Linear Placement. In: Proceedings of Design Automation Conference, Las Vegas, Nevada, pp. 50–56 (1972) 19. Shahookar, K., Mazumder, P.: VLSI Cell Placement Techniques. ACM Computing Surveys 23(2), 143–220 (1991) 20. Sun, W., Sechen, C.: Efficient and Effective Placement for Very Large Circuits. IEEE Transactions on Computer Aided Design of Integrated Circuits and Systems, 349–359 (1995) 21. Yang, Z., Areibi, S.: Global Placement Techniques for VLSI Physical Design Automation. In: 15th International Conference on Computer Applications in Industry and Engineering, San Diego, California (2002)

Bacterial Foraging Approach to Economic Load Dispatch Problem with Non Convex Cost Function B. Padmanabhan1, R.S. Sivakumar2, J. Jasper3, and T. Aruldoss Albert Victoire4 1

Power systems Engineering Power systems Engineering 3 Department of Electrical Engineering, Anna University of Technology, Coimbatore 4 Department of Electrical Engineering, Anna University of Technology, Coimbatore 2

Abstract. This paper presents a bacterial foraging-based optimization (BFBO) technique to solve non-convex economic load dispatch (NCELD) problem of thermal plants. The presented methodology can take care of economic dispatch problems involving constraints such as transmission losses, valve point loading, ramp rate limits and prohibited operating zones. The idea of BFBO is motivated by the natural selection which tends to eliminate the animals with poor foraging strategies and favour those having successful foraging strategies. The BFBO method is tested with two power system cases consisting of 6 and 13 thermal units. Comparison with similar approaches including Genetic Algorithm (GA), particle swarm optimization (PSO) and other versions of differential evolution (DE) are given. The presented method outperforms other state-of-the-art algorithms in solving economic load dispatch problems with the valve-point effect. Keywords: Bacterial foraging, Economic load dispatch, Non convex cost function, valve point effect.

1

Introduction

The objective of the economic load problem (ELD) is to schedule the committed generating unit outputs so as to meet the required load demand at minimum operating cost while satisfying all unit and system equality and inequality constraints [1]. In traditional EDPs, the cost function of each generator is approximately represented by a simple quadratic function and is solved using mathematical programming[2] based on several optimization techniques, such as dynamic programming [3], linear programming[4], homogenous linear programming[5], and nonlinear programming technique[6], [7]. However, real input–output characteristics display higher-order nonlinearities and discontinuities. Power plants usually have multiple valves that are used to control the power output of the unit. When steam admission valves in thermal units are first opened, a sudden increase in losses is observed. This leads to ripples in the cost function, which is known as the valve-point loading. The ELD problem with valve-point effects is represented as a non-smooth optimization [11]. . B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 577–584, 2011. © Springer-Verlag Berlin Heidelberg 2011

578

B. Padmanabhan et al.

The traditional algorithms can solve the ELD problems effectively only if the incremental fuel-cost curves of the generating units are monotonically increasing piece-wise linear functions. But, a practical ELD must include ramp rate limits, prohibited operating zones, valve-point effects and multi-fuel options. The resultant ELD is a challenging nonconvex optimization problem, which is hard to solve by the traditional methods. Recently, as an alternative to the conventional mathematical approaches, modern heuristic optimization techniques such as simulated annealing, evolutionary algorithms (EAs) (genetic algorithm, evolutionary programming, and evolution strategies) [8], particle swarm optimization, neural networks, and taboo search[9]-[12] have been given much attention by many researchers due to their ability to find an almost global optimal solution. These methods have drawbacks such as premature convergence and after some generations the population diversity would be greatly reduced. Inspired from the mechanism of the survival of bacteria, e.g., E. coli, an optimization algorithm, called Bacterial Foraging Algorithm (BFA) [10], has been developed. Chemotaxis, reproduction and dispersion are the three processes with the help of which global searching capability of this algorithm has been achieved. These properties have helped BFA to be applied successfully in several kinds of power system optimization problems especially in solving the NCELD problem. This paper considers two types of non-convex ELD problems, namely, ELD with prohibited operating zones and ramp rate limits (ELDPOZRR), ELD with valve-point loading effects (ELDVPL). The performance of the proposed method in terms of solution quality and computational efficiency has been compared with NPSO-LRS, CDE-QP and other techniques with non convex cost function.

2

Problem Formulation

The objective of the economic dispatch is to minimize the total generation cost of a power system over some appropriate period while satisfying various constraints. The objective function can be formulated as,

FT = min

 F ( P ) = min  ( a i∈ Ψ

i

i

i∈ Ψ

i

+ b i Pi + c i Pi 2 )

(1)

where FT is the total fuel cost of all generating units, i is the index of dispatchable units; Fi(Pi) is the cost function of the unit i, Pi is the power output of the unit i, Ψ is the set of all dispatchable units and ai , bi , ci are the fuel cost coefficients of the unit i.The general PED problem consists in minimizing FT subject to following constraints, Power Balance Constraint:



i∈ Ψ

The transmission Loss

Pi − P D − P L = 0

PL may be expressed using B-coefficients as,

(2)

Bacterial Foraging Approach to Economic Load Dispatch Problem

PL =

  PB i∈ Ψ j∈ Ψ

i

ij

Pj +

B i∈ Ψ

0i

Pi + B 00

579

(3)

PD is the total load demand; PL is the power losses and Bij is the power

where

loss coefficient. Generator Capacity Constraints: The power generated by each unit lies within their lower limit Pi limit Pi

max

min

and upper

. So that

Pi min ≤ Pi ≤ Pi max 2.1

(4)

ELDVPL

To consider the valve-point effects in the cost model, the rectified sinusoidal function should be incorporated into the quadratic function and the objective function min ( FT ) is represented by a more complex formula along with (2), (3) and (4) as,

Fi ( Pi ) = ai Pi 2 + bi Pi + ci + ei sin( f i ( Pi min − Pi ))

(5)

ei and f i are the fuel cost coefficients of generator i reflecting valve point

where effects. 2.2

ELDRRPOZ

The operating range of units is restricted by their ramp rate limits during each dispatch period. Consequently the power output of a practical generator cannot be varied instantaneously beyond the range along with (2), (3), (4) and (5) as it is shown in the following expression: As generation increases,

Pi − Pio ≤ URi

(6)

As generation decreases

Pio − Pi ≤ DRi

(7)

and

max ( Pi min , Pio − DR i ) ≤ Pi ≤ min( Pi max , Pio + UR i )

(8)

Mathematically, the feasible operating zones of unit can be described as follows:

Pi min ≤ Pi ≤ Pi ,l1 or Pi ,uj −1 ≤ Pi ≤ Pi ,l j , j = 2,..., ni or Pi ,uni ≤ Pi ≤ Pi max ,∀i ∈ψ

(9)

580

where

B. Padmanabhan et al.

Pi l, j is the lower bound of the prohibited zone j of unit i, Pi ,uj is the upper

bound of the prohibited zone j of unit i, i,

ni be the number of prohibited zones in unit

Pi min is the minimum generation limit of unit i and Pi max is the maximum

generation limit of unit i.

3

Bacterial Foraging: Review

BFO method was invented by Kevin M. Passino motivated by the natural selection which tends to eliminate the animals with poor foraging strategies and favour those having successful foraging strategies. The foraging strategy are given 3.1

Chemotaxis

Chemotaxis process is the characteristics of movement of bacteria in search of food and consists of two processes namely swimming and tumbling. A bacterium is said to be 'swimming' if it moves in a predefined direction, and 'tumbling' if moving in an altogether different direction. Suppose

θ i ( j , k , l ) represents

i

th

bacterium at j

th

chemotactic, k th reproductive and l th elimination dispersal step. C (i ) is the size of the step taken in the random direction specified by the tumble (run length unit). Then in computational chemotaxis the movement of the bacterium may be represented by

Δ (i )

θ i ( j + 1, k , l ) = θ i ( j , k , l ) + C (i)

(10)

Δ (i ) Δ (i ) T

where ∆ indicates a vector in the random direction whose elements lie in [–1, 1]. 3.2

Swarming

It is always desired that the bacterium which has searched optimum path of food search should try to attract other bacteria so that they reach the desired that the bacterium which has searched optimum path of food search should try to attract other bacteria so that they reach the desired place more rapidly. Swarming makes the bacteria congregate into groups and hence move a concentric pattern of groups with high bacterial density. Mathematically, swarming can be represented by S

p

i =1

m =1

J CC (θ , P ( j , k , l )) =  [ − d attract exp ( −ω attract  ( θ m − θ mi ) 2 )] + S

[ hrepellant exp (−ωrepellant i =1

where,

p

 (θ m =1

m

− θ mi ) 2 ) ]

(11)

J CC is the penalty added to the original cost function. J CC is basically the

relative distances of each bacterium from the fittest bacterium. S is the number of

Bacterial Foraging Approach to Economic Load Dispatch Problem

bacterium, ‘p’ represents number of parameters to be optimized, the fittest bacterium,

θm

581

is the position of

d attract , hrepellant , ωattract and ωrepellant are different

coefficients. 3.3

Reproduction

In reproduction, population members who have had sufficient nutrients will reproduce and the least healthy bacteria will die. The healthier half of the population replaces with the other half of bacteria which gets eliminated, owing to their poorer foraging abilities. This makes the population of bacteria constant in the evolution process. 3.4

Elimination and Dispersal

A sudden unforeseen event may drastically alter the evolution and may cause the elimination and/or dispersion to a new environment. They have the effect of possibly destroying the chemotactic progress, but they also have the effect of assisting in chemotaxis, since dispersal may place bacteria near good food sources. Elimination and dispersal helps in reducing the behavior of stagnation i.e. being trapped in a premature solution point or local optima.

4

Adaptive Strategy

A variation to the BFBO is given by employing adaptive strategies to improve the performance of the BFBO which controls the exploration of the whole search space and the exploitation of the promising areas. This strategy is used to analyze the run length parameter of the BFBO. This improvement is achieved by enabling the bacterial foraging algorithm to adjust the run-length unit parameter dynamically during algorithm execution in order to balance the exploration/exploitation tradeoff. Living in groups allows individuals to allocate foraging effort between two different roles, namely, the producer and the scrounger. The “producer” can be used to locate food patches independently while the “scrounger” can be used to exploit the food discovered by other group members. The bacterial producers explore the search space and have the responsibility to find the promising domains and to leave the local optima that have been visited while the bacterial scroungers focus on the precision of the found solutions. The producer-scrounger foraging is used to dynamically determine the chemotactic step sizes for the whole bacterial colony during a run, hence dividing the foraging procedure of artificial bacteria colony into multiple explore and exploit phases. 4.1

Pseudocode If (t mod n = 0) then If ( f best < ϵ (t) ) then C (t + 1) = C (t - n) / α є (t + 1)= є (t) / β ELSE

582

B. Padmanabhan et al.

C ( t+1 )=C (t-n); є (t + 1)=є (t - n); END IF ELSE C (t + 1)=C (t) є (t + 1)=є (t); END IF where t is the current generation number, fbest is the best fitness value among all the bacteria in the colony, ε(t) is the required precision in the current generation, and n, α, and β are user-defined constants.

5

Simulation Result

To validate the effectiveness of the proposed method, two test cases such as 6 unit and 13 unit systems with non convex cost function are taken. The result obtained from proposed method has been compared with CDE-QP [13] for 13 generator system; with MPSO and other techniques for 6 generator system. The software has been written in MATLAB-R2009a language and executed on a 2.0-GHz Pentium Dual personal computer with 1400-MB RAM. Case Study 1 A system with six generators with ramp rate limit and prohibited operating zone is used here and has a total load of 1263 MW. The input data have been adopted from [13]. Results obtained from DE, BFBO, PSO and new coding-based modified PSO [12] and other methods have been presented here. Table 2 shows the frequency of convergence in 50 trial runs. It is clear from Table 1 shows that the proposed method produces a much better solution compared to the MPSO, NPSO-LRS, IDE and other methods. Table 1. Comparison of cost among different methods for 50 trials (6-unit system)

Method GA PSO NPSO – LRS MPSO IDE BFBO

Generation Cost ($/h) Maximum cost Minimum cost Average cost 15,524 15,459 15,469 15,492 15,450 15,454 15,452 15,450 15,450.5 15,447 15,455 15,447 15,359 15,351 15,356 15, 352 15,348 15,350

Table 2. Frequency of convergence for 50 trials (6-unit system)

Methods BFBO

Range of Generation Cost ($/h) 15000-15350 15350-15400 15400-15500 32 10 8

Bacterial Foraging Approach to Economic Load Dispatch Problem

583

Case Study 2 This test case is a NSELD of 13 units with valve point loading and has a load demand of 1800MW.The input data are given in [13]. The result obtained from presented method has been compared with CDE-QP [13] and other methods. Table 4 shows the frequency of convergence in 50 trial runs. It is clear from Table 3 and 4 that the proposed method produces a much better solution with less computation time compared to the ICA-PSO, CDE-QP and other methods. Table 3. Comparison of cost among different methods for 50 trials (13-unit system)

Method PS STDE MDE ICA-PSO CDE-QP BFBO

Generation Cost ($/h) Maximum cost Minimum cost Average cost 18404.04 18048.21 18190.32 18128.87 17963.89 18046.38 17969.09 17960.39 17967.19 17978.14 17960.37 17967.94 17944.81 17938.95 17943.133 17933.61 17918.73 17921.12

Table 4. Frequency of convergence for 50 trials (6-unit system)

Methods BFBO

6

Range of Generation Cost ($/h) 17800-17920 17920-17930 17930-17940 28 12 10

Conclusion

This paper presents new combined approaches combining to solve the ELD problems of electric energy with the valve-point effect. The BFBO algorithm has the ability to find the better quality solution and has better convergence characteristics, computational efficiency, and robustness. It is clear from the results obtained by different trials that the proposed BFBO method has good convergence property and can avoid the shortcoming of premature convergence of other optimization techniques to obtain better quality solution. Two case studies have been used and the simulation results indicate that this optimization method is very accurate and converges very rapidly so that it can be used in the practical optimization problems. Due to these properties, the BFBO method in the future can be tried for solution of large unit systems and dynamic ELD problems in the search of better quality results.

References [1] Abido, M.A.: A novel multiobjective evolutionary algorithm for environmental/ economical power dispatch. Elect. Power Syst. Res. 65, 71–81 (2003) [2] Wood, A.J., Wollenberg, B.F.: Power Generation, Operation and Control. Wiley, New York (1994)

584

B. Padmanabhan et al.

[3] Lin, C.E., Viviani, G.L.: Hierarchical economic dispatch for piecewise quadratic cost functions. IEEE Trans. Power App. Syst. PAS-103(6), 1170–1175 (1984) [4] Yang, H.T., Chen, S.L.: Incorporating a multi-criteria decision procedure into the combined dynamic programming/production simulation algorithm for generation expansion planning. IEEE Trans. Power Syst. 4(1), 165–175 (1989) [5] Granville, S.: Optimal reactive dispatch through interior point methods. In: Proc. IEEE Summer Meeting, Seattle, WA, Paper no. 92 SM 416-8 PWRS (1992) [6] Liang, Z.X., Glover, J.D.: A zoom feature for a programming solution to economic dispatch including transmission losses. IEEE Trans. Power Syst. 7(3), 544–550 (1992) [7] Chen, C.L., Wang, C.L.: Branch-and-bound scheduling for thermal generating units. IEEE Trans. Energy Convers. 8(2), 184–189 (1993) [8] Park, J.-B., Lee, K.-S., Shin, J.-R., Lee, K.Y.: A particle swarm optimization for economic dispatch with nonsmooth cost function. IEEE Trans. Power Syst. 20(1), 34–42 (2005) [9] Kumar, J., Seblé, G.B.: Clamped state solution of artificial neural network for real-time economic dispatch. IEEE Trans. Power Syst. 10(2), 925–931 (1995) [10] Sinha, N., Chakrabarti, R., Chattopadhyay, P.K.: Evolutionary programming techniques for economic load dispatch. IEEE Trans. Evol. Comput. 7(1), 83–94 (2003) [11] Victoire, T.A.A., Jeyakumar, A.E.: Hybrid PSO-SQP for economic dispatch with valvepoint effect. Elect. Power Syst. Res. 71(1), 51–59 (2004) [12] Walters, D.C., Sheble, G.B.: Genetic algorithm solution of economic dispatch with valve point loading. IEEE Trans. Power Syst. 8(3), 1325–1332 (1993) [13] Panigrahi, B.K., Yadav, S.R., Agrawal, S., Tiwari, M.K.: A clonal algorithm to solve economic load dispatch. Electric Power System Research 77, 1381–1389 (2007)

Static/Dynamic Environmental Economic Dispatch Employing Chaotic Micro Bacterial Foraging Algorithm Nicole Pandit1, Anshul Tripathi1, Shashikala Tapaswi1, and Manjaree Pandit2 1

2

ABV-IIITM, Gwalior, India Department of Electrical Engineering, MITS, Gwalior, India {nicole0831,anshul.iiitm1}@gmail.com, [email protected], [email protected]

Abstract. Environmental Economic Dispatch is carried out in the energy control center to find the optimal thermal generation schedule such that power balance criterion and unit operating limits are satisfied and the fuel cost as well as emission is minimized. Environmental economic dispatch presents a complex, dynamic, non-linear and discontinuous optimization problem for the power system operator. It is quite well known that gradient based methods cannot work for discontinuous or nonconvex functions as these functions are not continuously differentiable As a result, evolutionary methods are increasingly being proposed. This paper proposes a chaotic micro bacterial foraging algorithm (CMBFA) employing a time-varying chemotactic step size in micro BFA. The convergence characteristic, speed, and solution quality of CMBFA is found to be significantly better than classical BFA for a 3-unit system and the standard IEEE 30-bus test system.

1

Introduction

Bacterial Foraging Algorithm (BFA), proposed by Passino [1], is a new evolutionary optimization algorithm inspired by the social foraging behavior of E. coli bacteria found in human intestines. The biology behind the foraging strategy of E. coli is imitated and formulated into a powerful optimization algorithm. The bacteria search for nutrition with the help of two basic operations i.e. swim and tumble, together known as chemotaxis. However, the classical BFA is found to have certain limitations like slow convergence and saturation, particularly for large dimension problems. To handle these issues many improved versions of the classical BFA have been proposed. In reference [2] a micro-bacterial foraging algorithm is presented which evolves with a very small population. Ref [3] proposed a hybrid algorithm which combines the PSO with BFA to improve the efficiency and accuracy of the classical BFA. Environmental economic dispatch (EED) is a highly non linear, discontinuous, non differentiable problem where objective function may have multiple local minima. Reference [4] summarizes the various algorithms for solving environmental dispatch problem. In [5] a method which combines cost and emission objectives into a single function using price penalty factor is proposed. The environmental economic dispatch problem has been solved using non dominated sorting GA [6], PSO based methods B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 585–592, 2011. © Springer-Verlag Berlin Heidelberg 2011

586

N. Pandit et al.

[7] and bacterial foraging algorithm [8]. Most researchers have concentrated only on the static environmental economic dispatch problem (SEED) except ref. [6] where dynamic EED (DEED) has been implemented. This paper proposes a modified BFA by integrating a micro bacterial foraging algorithm [2] with chaotically varying chemotactic step size. The idea is to (i) reduce the time of operation by reducing bacteria count and (ii) to improve exploration and exploitation by employing a time varying step size. The results of the CMBFA are compared with the classical BFA and with results from literature, and are found to be superior.

2

Static/Dynamic Combined Environmental Economic Dispatch

The objective of static environmental economic dispatch (SEED) problem is to determine the generated powers Pi of units for a load of PD so that the total fuel cost, FT and total emission content ET ,expressed below are simultaneously minimized. N

N

i =1

i =1

FT =  Fi ( Pi ) =  ( ai Pi + bi Pi + c i )$ / h

(1)

2

where ai ,bi and ci are the fuel-cost coefficients, α i , β i , γ i , ξ i are the pollution coefficients and Pi is the output of the ithunit. Emission content of sulfur oxides (SOx) and nitrogen oxides (NOX) caused by fossil plants can be represented in ton/h as N

N

i =1

i =1

ET =  Ei ( Pi ) = 10 − 2 * (α i + β i Pi + γ i Pi 2 ) + ξ i exp(λi Pi )

(2)

In combined EED formulation, the two objective problem is converted into a single objective problem using the price penalty factor, pf which blends the emission costs max

with the fuel cost and for ith unit it is defined as Fi (P i

max

)/ Ei (P i

) [5]. The pfi

values are arranged in ascending order; the maximum capacity of each unit,

(P ) max

i

is added one at a time, starting from the smallest pfi unit, until total demand PD is met When the above condition is met, the pfi associated with the last unit in the process is the price penalty factor pf ($/ton) for the given load PD. Then equation (3) can be solved to obtain environmental economic dispatch. Minimize Z=

N

N

i =1

i =1

(ai Pi2 + bi Pi + ci ) + pf *(10−2 *(αi + βi Pi +γ i Pi2 ))+ξi exp(λi Pi ) $/h (3)

Subject to Constraints i) Unit operating limits constraint

Pimin ≤ Pi ≤ Pi max

i =1,2,...,N

(4)

Static/Dynamic Environmental Economic Dispatch Employing CMBFA

587

ii) Power balance constraint N

 P − (P i

i =1

D

+ PL ) = 0

(5)

The transmission losses are expressed as N

PL =  i =1

N

PB i

j =1

N

ij

P j +  B oi Pi + B oo

(6)

i =1

iii) Unit ramp-rate limit constraints When the generator ramp rate limits are considered, the operating limits are modified as follows:

Max ( Pi min , Pi o − DRi ) ≤ Pi ≤ Min( Pi max , Pi o + URi )

(7)

The previous operating point of ith generator is Pio and DRi and URi are the down and up ramp rate limits respectively. iv) Prohibited operating zones The cost curves of practical generators are discontinuous as whole of the unit operating range is not always available for allocation and can be expressed as follows:

Pi

   ∈    

Pi

min

P ikU −

1

.......... P

U izi

≤

Pi ≤

P i 1L

≤

Pi ≤

P ik L

.......... ≤

Pi ≤

(8)

... Pi

max

Here zi are the number of prohibited zones in i generator curve, PikL is the lower th

limit and PikU− 1 is the upper limit of kth prohibited zone of ith generator. 2.1

Dynamic Economic Emission Dispatch

Dynamic economic emission dispatch (DEED) deals with sharing the system load including system losses among the available generators in such a way that all equality and inequality constraints mentioned above are met and the cost as well as emission are minimized for each time interval. The combined dynamic economic emission dispatch (DEED) problem is formulated for each time interval as defined in eq. (3).

3

Classical BFA Algorithm

The bacterial foraging algorithm (BFA) was introduced by Kevin M. Passino [1] motivated by the group foraging strategy of a swarm of E. coli bacteria which search for nutrients so as to maximize energy obtained per unit time. The process, by which a bacterium moves by taking small steps while searching for nutrients, is called chemotaxis. The main idea of BFA is imitating chemotactic movement of virtual

588

N. Pandit et al.

bacteria in the problem search space. The classical BFA algorithm consists of four main steps, wiz. chemotaxis, swarming, reproduction, and elimination and dispersal. The movement of the bacterium may be represented as:

θ i ( j + 1, k, l ) = θ i ( j, k, l ) + C(i)φ( j) Where

θ i ( j , k , l ) is

φ( j) =

where

Δ(i)

(9)

ΔT (i)Δ(i)

the position of the ith bacterium at jth chemotactic

kth

reproductive, and lth elimination and dispersal step. C(i) is the size of the step taken in the random direction of tumble. Here, Δ indicates a movement in arbitrary direction whose elements lie in [-1, 1]. 3.1

Chaotic Micro Bacterial Foraging Algorithm (CMBFA)

In micro BFA [2] a 3-bacteria population searches iteratively for large dimension complex domain. A chaotically varying chemotactic step size is employed to update the position of the bacteria. The best bacterium retains its position. The second best bacterium is moved to a position very close to the best one, the least fit bacterium is killed and its place is taken by a new bacterium. Reproduction is not carried out as the bacteria population is very small. Elimination dispersion step of original algorithm is also retained. The step size C in eq.(9) is changed chaotically making use of one of the dynamic systems called the logistic map.

C (t ) = μ × C (t − 1) × [1 − C (t − 1)]

(10)

Here, t is the iteration count and µ is a parameter, 0 ≤ µ ≤ 4 which controls the variation of the chaotic sequence. The step size C is iteratively varied. The system at (10) displays chaotic behavior when µ=4 and C (0) ∉ 0,0.25,0.5,0.75,1.0 . Fig. 1

{

1

}

Step length

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

Iteration 5

10

15

20

25

30

35

40

Fig. 1. Step length in CMBFA for µ =4, C(t=0) =0.64

45

50

Static/Dynamic Environmental Economic Dispatch Employing CMBFA

589

shows the variation of step length for C(0)=0.64. For C(0) =(0.0, 0.25,0.5,0.75, 1.0) there will be no variation in step length with time..

4

Results and Discussion

The performance of CMBFA is compared for test systems having i) Prohibited operating zones(POZ) ii) ramp rate limits (RRL)and iii) transmission losses in addition to the iv)generating capacity constraints and v) power balance equation. The optimal solutions are computed for i) Test case I: SED for three thermal generating units system [9] with POZ and RRL without losses ii) Test Case II: Test case I with transmission losses iii) Test Case III: IEEE 30-bus test system for SEED problem [7]. iv)Test Case IV: The DEED for Test Case III for 24-hour load variation. Simulations were carried out using MATLAB 7.0.1 on a Pentium IV processor, 2.8 GHz. with 1 GB RAM. The RRL is assumed as [.08,.11,.15,.18,.15,.18]. 4.1

Parameter Setup and Effect of Bacteria Population

For both BFA and CMBFA, the number of chemotactic steps is taken as 50, length of swimming loop and elimination dispersal loop are both set at 4 and number of iterations are 100. For the Test case I base load was 300 MW. Results for different bacteria counts are tabulated in Table 1. The standard deviation (S.D) improves with bacteria count. 3498 BFA CMBFA

3496 3494

cost

3492 3490 3488 3486 3484 3482

0

10

20

30

40

50 iteration

60

70

80

90

100

Fig. 2. Comparison of convergence characteristics of BFA and CMBFA for test case I Table 1. Effect of change in bacteria count on BFA performance: Test Case I (out of ten trials)

Bacteria count Minimum Cost($/hr) Maximum Cost($/hr) 4 3482.86797 3484.91036 6 3482.86749 3482.87451 8 3482.86798 3482.88374 10 3482.86796 3484.90744 12 3482.86760 3482.86970

Mean 3483.1 3482.9 3482.9 3483.1 3482.9

SD 0.3666 0.0020 0.0013 0.00046 0.00026

590

N. Pandit et al.

The CPU time required increases with bacteria count. The convergence behavior of BFA and CMBFA is plotted in Fig. 2. The CMBFA converges faster with CPU time 0.1235s/iteration as compared to BFA with 0.2454s/iteration. The performance of CMBFA is compared with GA[9] APSO[10], AIS[11] and HBBO[12] in Table 2. The results of CMBFA are superior to others as minimum cost at zero constraint violation is achieved. In [10] and [12] the ramp rate limits for unit three are violated and in [9] and [11] the constraint violation (shown bold) is high which makes the result infeasible. Table 2. Comparison of best results for Test Case II

Output(MW) CMBFA BFA HBBO[12] APSO[10] P1 200.4637 205.88944 207.637 200.528 P2 78.3750 73.06689 87.2833 78.2776 P3 34.0000 34.0000 15.0000 33.9918 Total power 312.8387 312.9563 309.9203 312.7974 loss 12.8387 12.9572 9.9204 12.8364 ∑Pi -PD-PL 0.0 -0.0009 -0.0001 -0.039 cost ($/h) 3634.7691 3635.1583 3619.7565 3634.3127 4.2

AIS[11 ] 198.7575 77.99225 35.19886 311.9486 11.816 0.1326 3624.44

GA[ 9] 194.26 50 79.62 323.88 24.011 -0.131 3737.20

Static/Dynamic Environmental Economic Dispatch (SEED/DEED)

Table 3 compares BFA and CMBFA for best cost and best emission cases. The results of CMBFA are significantly better for both cases. To get the best compromise solution, the price penalty factor pf helps in obtaining a solution which gives equal weight to minimization of cost as well as emission. In Table 4, for Load 2.4 MW to 2.8 MW, pf=4470.2746 and for larger values of Load, pf=5928.7134. Table 3. Comparison of best cost and emission solutions by BFA and CMBFA for Test Case III Variables P1(per unit) P2 (per unit) P3 (per unit) P4 (per unit) P5 (per unit) P6 (per unit) COST(&/h) EMISSION(ton/h) LOSS(MW/h)

∑Pi -PD-PL

BEST COST SOLUTION BFA CMBFA 0.10422553 0.1209 0.28841535 0.2812 0.58869564 0.6050 0.94961114 0.9801 0.54599832 0.5239 0.38274581 0.3473 606.34386659 605.9438 0.22571781 0.2484 0.02540773 0.0248 0.00028407 -0.0005

BESTEMISSION SOLUTION BFA CMBFA 0.40409088 0.3953 0.38769951 0.4304 0.48109197 0.5386 0.46402481 0.4238 0.65923582 0.5584 0.47191157 0.5214 637.61844944 641.5610 0.19512784 0.1937 0.03407091 0.0339 -0.00001635 -0.0000

Static/Dynamic Environmental Economic Dispatch Employing CMBFA

591

Table 4. Best Compromise Solution for DEED using CMBFA for IEEE 30-bus system Hour H1 H2 H3 H4 H5 H6 H7 H8 H9 H10 H11 H12 H13 H14 H15 H16 H17 H18 H19 H20 H21 H22 H23 H24

5

P1 0.1195 0.2195 0.2195 0.2195 0.1195 0.2195 0.2195 0.2195 0.2165 0.2123 0.2195 0.2195 0.2195 0.2195 0.2195 0.2195 0.2195 0.2195 0.2195 0.2195 0.2195 0.2195 0.2195 0.2195

P2 0.5013 0.4582 0.3613 0.3613 0.5013 0.5013 0.5013 0.5013 0.5013 0.4881 0.4000 0.5013 0.5013 0.4706 0.5013 0.5013 0.5013 0.5013 0.3613 0.4623 0.3613 0.3613 0.3993 0.3613

P3 0.5483 0.5483 0.3683 0.3683 0.5483 0.3683 0.4559 0.5483 0.5483 0.5052 0.5483 0.5483 0.5483 0.5483 0.4200 0.5279 0.4460 0.5483 0.4250 0.4583 0.5483 0.5271 0.4255 0.3683

P4 0.4284 0.4930 0.6284 0.5149 0.5332 0.6284 0.6284 0.6284 0.6284 0.5288 0.6284 0.6284 0.5498 0.4921 0.5950 0.4284 0.6284 0.6284 0.5459 0.4799 0.4284 0.4284 0.4933 0.5697

P5 0.3586 0.3586 0.5386 0.5386 0.5386 0.4624 0.5386 0.4953 0.5386 0.5386 0.5386 0.5386 0.5386 0.5386 0.5132 0.5386 0.5386 0.3942 0.5386 0.4499 0.5386 0.5386 0.4307 0.3586

P6 0.4682 0.3946 0.4078 0.5946 0.3946 0.5115 0.3946 0.3946 0.3946 0.5920 0.5946 0.5256 0.5946 0.5946 0.5840 0.5946 0.3946 0.3946 0.5452 0.5266 0.4240 0.3946 0.4644 0.4854

Cost 537.0861 541.6875 543.1863 570.7961 575.0908 586.9743 594.2952 605.0353 613.9105 632.0021 638.2734 646.3690 651.0041 633.4414 622.1056 627.1872 592.1582 583.2600 574.8516 572.2063 553.2874 541.7397 531.6621 512.8081

Emission 0.2014 0.1993 0.2018 0.1997 0.2014 0.2004 0.1999 0.1997 0.1996 0.1975 0.2000 0.2169 0.1977 0.1974 0.1996 0.1966 0.2003 0.2012 0.2003 0.1984 0.1977 0.1979 0.1984 0.2018

LOSS 0.0242 0.0222 0.0240 0.0273 0.0256 0.0313 0.0283 0.0273 0.0278 0.0311 0.0293 0.0313 0.0321 0.0298 0.0332 0.0303 0.0284 0.0262 0.0257 0.0266 0.0202 0.0196 0.0226 0.0227

LOAD 2.4 2.45 2.5 2.57 2.61 2.66 2.71 2.76 2.8 2.834 2.9 2.93 2.92 2.834 2.8 2.78 2.7 2.66 2.61 2.57 2.5 2.45 2.41 2.34

Conclusions

A chaotic micro BFA (CMBFA) is developed in this paper and its performance is compared with the classical BFA for four test cases having complex, discontinuous and non-linear objective functions and constraints. The test results clearly show that i) the consistency of classical BFA depends on bacteria count. The mean cost and S.D improved with bacteria count. ii) The CMBFA is found to be many times faster.iii) The CMBFA has got a superior performance as compared to classical BFA. iv) BFA and CMBFA both produce feasible solutions with full constraint satisfaction but CMBFA produced lower cost, emission and losses as compared to BFA.

References 1. Passino, K.M.: Biomimicry of bacterial foraging for distributed optimization and control. IEEE Control Syst. Mag. 22(3), 52–67 (2002) 2. Dasgupta, S., Biswas, A., Das, S., Panigrahi, B.K., Abraham, A.: A Micro-Bacterial Foraging Algorithm for High-Dimensional Optimization. In: IEEE Congress on Evolutionary Computation, pp. 785–792 (2009) 3. Long, L.X., Jun, L.R., Ping, Y.: Bacterial Foraging Global Optimization Algorithm Based On the Particle Swarm Optimization. In: IEEE International Conference on Intelligent Computing and Intelligent Systems, pp. 22–27 (2010)

592

N. Pandit et al.

4. Talaq, J.H., El-Hawary, F., El-Hawary, M.E.: A summary of environmental and economic dispatch algorithms. IEEE Trans. Power Syst. 9(3), 1508–1516 (1994) 5. Palanichamy, C., Srikrishna, K.: Economic Thermal Power Dispatch with Emission Constraint. Journal of the Institution of Engineers (India) 72, 11–18 (1991) 6. Basu, M.: Dynamic economic emission dispatch using nondominated sorting genetic algorithm-II. Electrical Power and Energy Systems 30, 140–149 (2008) 7. Abido, M.A.: Multiobjective particle swarm optimization for environmental/economic dispatch problem. Electric Power Systems Research 79, 1105–1113 (2009) 8. Hota, P.K., Barisal, A.K., Chakrabarti, R.: Economic emission load dispatch through fuzzy based bacterial foraging algorithm. Electrical Power and Energy Systems 32, 794–803 (2010) 9. Chen, P.H., Chang, H.C.: Large scale economic dispatch approach by genetic algorithm. IEEE Transactions on Power Systems 10(4), 1919–1926 (1995) 10. Panigrah, B.K., Ravikumar, P.V., Das, S.: An Adaptive Particle Swarm Optimization Approach for Static and Dynamic Economic Load Dispatch. International Journal on Energy Conversion and Management 49, 1407–1415 (2008) 11. Panigrahi, B.K., Yadav, S.R., Agrawal, S., Tiwari, M.K.: A clonal algorithm to solve economic load dispatch. Electric Power System Research 77, 1381–1389 (2007) 12. Bhattacharya, A., Chattopadhyay, P.K.: Hybrid differential evolution with biogeographybased optimization for solution of economic load dispatch. IEEE Transactions on Power System 25(4), 1955–1964 (2010)

Artificial Bee Colony Algorithm with Self Adaptive Colony Size Tarun Kumar Sharma1, Millie Pant1, and V.P. Singh2 1

IIT Roorkee MIT Saharanpur {taruniitr1,millidma,singhvp3}@gmail.com 2

Abstract. The Artificial Bee Colony or ABC is one of the newest additions to the class of Nature Inspired Algorithms (NIA) mimicking the foraging behavior of honey bees. In ABC, the food locations represent the potential candidate solution. In the present study an attempt is made to generate the population of food sources (Colony Size) adaptively with the help of proposed algorithm ABC-SAC (artificial bee colony with self-adaptive colony). ABC-SAC is further is modified by varying the behavior of bees in search of food and the corresponding variants are named as ABC-SAC1 and ABC-SAC2. The proposed algorithms are validated on a set of standard benchmark problems with varying dimensions taken from literature and the numerical results are compared with the basic ABC and one its recent variant gbest ABC, which indicate the competence of the proposed algorithms. Keywords: Artificial Bee Colony, Self Adaptive, Foraging behaviour, Optimization.

1

Introduction

Artificial Bee Colony (ABC) optimization algorithm is a population-based swarm intelligence algorithm that was originally proposed by Karaboga, Erciyes University of Turkey in 2005 [1] [2]. It simulates the foraging behaviour that a swarm of bees perform. In this algorithm there are three groups of bees, the employed bees (bees that determines the food source (possible solutions) from a prespecified set of food sources and share this information (waggle dance) with the other bees in the hive), the onlookers bees (gets the information of food sources from the employed bees in the hive and select one of the food source to gathers the nectar) and the scout bees (responsible for finding new food sources). A brief overview of the algorithm is given in section 2. Like other population based search algorithm, ABC starts with a population of potential candidate solutions which in case of ABC represent the food locations (flower patch etc.). The amount of nectar in the food source represents its goodness (fitness value). In the basic ABC, the number of food sources is fixed in the beginning of the algorithm. However, practically speaking it is very rare that the every patch will contain the same number of flowers. Keeping this in mind, in the present study we propose a modified variant of ABC in which the population of food sources change adaptively. The corresponding algorithm is named ABC-SAC which B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 593–600, 2011. © Springer-Verlag Berlin Heidelberg 2011

594

T.K. Sharma, M. Pant, and V.P. Singh

is further modified into two variants ABC-SAC1 and ABC-SAC2. In ABC-SAC1, elitism is used where the bees are always guided towards the best food source (i.e. the one having the best fitness function value). In ABC-SAC2, both local and global explorations are kept into account. The remaining of the paper is organized as follows; in the next section we give a brief overview of the basic ABC algorithm. In section 3, the proposed variants are described. Numerical results are given in section 4 and finally the paper concludes with section 5.

2

Artificial Bee Colony Algorithm

ABC is one of the newest algorithms based on the foraging behavior of insects. It tries to model natural behavior of real honey bees in food foraging. Since ABC algorithm is simple in concept, easy to implement, and has fewer control parameters, it has been widely used in many fields. ABC algorithm has been applied successfully to a large number of various optimization problems [3-10]. The colony of artificial bees contains three groups of bees: employed bees, onlookers and scouts. A bee waiting on the dance area for making a decision to choose a food source is called onlooker and one going to the food source visited by it before is named employed bee. The other kind of bee is scout bee that carries out random search for discovering new sources. The position of a food source represents a possible solution to the optimization problem and the nectar amount of a food source corresponds to the quality (fitness) of the associated solution. In the algorithm, the first half of the colony consists of employed artificial bees and the second half constitutes the onlookers. The number of the employed bees or the onlooker bees is equal to the number of solutions in the population. At the first step, the ABC generates a randomly distributed initial population of NP solutions (food source positions), where NP denotes the size of population. Each solution xi where i =1, 2,..., NP is a D-dimensional vector, where D is the number of optimization parameters. After initialization, the population of the positions (solutions) is subjected to repeated cycles, C =1, 2,..., MCN of the search processes of the employed bees, the onlooker bees and scout bees. An employed bee produces a modification on the position (solution) in her memory depending on the local information (visual information) and tests the nectar amount (fitness value) of the new source (new solution). Provided that the nectar amount of the new one is higher than that of the previous one, the bee memorizes the new position and forgets the old one. Otherwise she keeps the position of the previous one in her memory. After all employed bees complete the search process; they share the nectar information of the food sources and their position information with the onlooker bees on the dance area. An onlooker bee evaluates the nectar information taken from all employed bees and chooses a food source with a probability related to its nectar amount. As in the case of the employed bee, she produces a modification on the position in her memory and checks the nectar amount of the candidate source. Providing that its nectar is higher than that of the previous one, the bee memorizes the new position and forgets the old one. An artificial onlooker bee chooses a food source depending on the probability value associated with that food source pi, calculated as Eq. (1): fit p i = NP i (1)  fit i i =1

Artificial Bee Colony Algorithm with Self Adaptive Colony Size

595

where fiti is the fitness value of the solution i which is proportional to the nectar amount of the food source in the position i and NP is the number of food sources which is equal to the number of employed bees. In order to produce a candidate food position from the old one in memory, the ABC uses the following Eq. (2):

vij = xij + φij ( xij − xkj )

(2)

where k ∈ {1, 2, …, NP} and j ∈ {1, 2, …, D} are randomly chosen indexes. Moreover, k ≠ i. Øij is a random number between [-1, 1]. It controls the production of neighbor food sources around xij and represents the comparison of two food positions visible to a bee. This can be seen from Eq. (2), as the difference between the parameters of the xij and xkj decreases, the perturbation on the position xij decreases, too. Thus, as the search approaches to the optimum solution in the search space, the step length is adaptively reduced. After each candidate source position is produced and evaluated by the artificial bee, its performance is compared with that of its old one. If the new food source has equal or better quality than the old source, the old one is replaced by the new one. Otherwise, the old one is retained. If a position cannot be improved further through a predetermined named “limit”, then that food source is assumed to be abandoned. The corresponding employed bee becomes a scout. The abandoned position will be replaced with a new food source found by the scout. Assume that the abandoned source xi, then the scout discovers a new food source to be replaced with xi. This operation can be defined as in Eq. (3): j j j xij = xmin + rand ()( x max − x min )

where

j j xmax and xmin are upper and lower bounds of parameter j, respectively.

3

Proposed ABC-SAC Algorithms and Its Variant

(3)

In the proposed ABC-SAC algorithm an attempt is to generate adaptive Colony Size. First of all we randomly generate some initial population of solutions, which at a later stage keep on changing adaptively. The Food Source position (NP) of subsequent generations is taken as the average of the population size attribute from all individuals in the current population as follows: Initialize the population of solutions xi,j,G. Then declare p as variable (Initialize p=0) for (i=0; i f it(Xi ) THEN Xi = Xi END FOR. 3. FOR each onlooker bee do Select food source Xi depending on probability Np−1

pi = f it(Xi )/

 j=0

/

Generate Xi as in equation (3) and (4). / / IF f it(Xi ) > f it(Xi ) THEN Xi = Xi END FOR.

f it(Xj )

(5)

614

A. Halder et al.

4. If any food source is abandoned by employed bees, re-initialize the food source by scout bee. Repeat steps (1), (2), (3) and (4) until required condition is reached.

4 4.1

Methodology Segmentation of the Lip-Contour

Several algorithms for the lip segmentation are available in the literature [2], [3]. In this paper, we, however, employ fuzzy k-means clustering algorithm to segment the lip region from the rest of the facial expression. In order to segment the lip region, we first localize the mouth region as shown in Fig. 3.a. Then a conversion from r-g-b to l-a-b color space is undertaken. The k-means clustering algorithm is applied next on this image to get three clusters. The three clusters are: skin, lip and teeth regions. The cluster with the highest intensity variance in l-a-b color space is declared as the lip region. Thus we select the lip cluster (Fig. 3.b) to determine the lip-contour.

Fig. 3. The segmented mouth region obtained by FCM algorithm

4.2

Parameter Extraction of a Given Lip-Contour Using Artificial Bee Colony Optimization Algorithm

Artificial Bee Colony optimization(ABC) algorithm proposed by Karaboga and Basturk [1] offers promising solution to a global optimization problem. In this paper, we employ artificial bee colony as the optimization algorithm to determine the lip parameters of a given subject carrying a definite emotion.Given a finite set of selected points on the lip boundary of a segmented mouth region, and a model lip curve, we need to match the response of the model curve with the selected data points by varying the parameters of the model curve. The set of parameters for which the best matching takes place between the model generated data points and the selected lip boundary points are the results of a system identification procedure adopted here. Let, y = f (x) be the model curve. Then for all (x, y) lying on the curve, we obtain G(x, y) = 1, and for all points y = f (x), G(x, y) = 0. Let, L(x, y) = 1 for all valid data points on the outer boundary of a segmented lip. We use a performance evaluation metric J, where   J= |G(x, y) − L(x, y)| (6) ∀x ∀y,y=f (x)

Emotion Recognition from the Lip-Contour of a Subject

615

In ABC algorithm, we used J as the fitness functions, where we wanted to minimize J for all valid (x, y) on the lip boundary. The ABC considers 9-parameter food sources, finds out the neighborhood food source by mutation operation, and selects the best of the searched food source and the original food source to determine the parameter vector in the next iteration. This is done in parallel for NP number of parameter vectors, where NP is the population size. The algorithm is terminated when the error limit, defined by the difference of J’s between the best of the previous and the current iteration is below a prescribed threshold. The best fit parameter vector is the parameter set of the best model lip-contour matched with a given lip boundary data points. 4.3

Emotion Classification from Measured Parameters of the Lip-Contour Model

It is noted from a large number of lip-contour instances that there exist at least two parameters of the lip model clearly distinctive of individual emotions. So, any typical machine learning/statistical classifier can be employed to classify the different emotional status from the parameter of lip-contour. In this paper, we use Linear Support Vector Machine (SVM) [2] classifier for emotion classification from the lip data. In our basic scheme, we employed five SVM networks, one each for joy, anger, sadness, fear and relaxation. The ith SVM network is trained with all of the training instances of the ith class with positive levels, and all other training instances with negative levels. The decision logic box driven by the five SVM networks ultimately recognizes the emotion corresponding to the supplied feature vector X. The decision logic works in the following manner. If only one input of the decision logic is +1, it infers the corresponding class number at the output. For, example, if the SVM-disgust only generates a +1, the output of the decision logic will be the class number for the emotion class: disgust. When more than one input of the decision logic is +1, the decision is taken in two steps. First, we count the number of positive instances falling in the small neighborhood of the given pattern for each emotion class with its corresponding SVM-output +1. Next, the emotion class with the highest count is declared as the winner. The decision logic thus takes decision based on the principle of “majority voting”, which is realized here by the count of positive instances in the neighborhood of the given test pattern. The “neighborhood” here is defined as a 9-dimensional sphere function around a given test pattern, considering it as the center of the sphere. The radius of the sphere is determined from the measurements of standard deviation in the individual features. The largest among the standard deviations for all the features is considered as the radius of the “neighborhood” in the data points, representing positive instances in a given emotion class. The radius of different emotion classes here thus is different. This, however, makes sense as the data density (per unit volume in 9-dimensional hyper space) for different emotion classes is non-uniform.

616

5

A. Halder et al.

Experiments and Results

The experiment has two phases. In the first phase, we determine weight vectors of 5 SVM classifiers, each one for one emotion class, including anger, disgust, fear, happiness and sadness. We had 50 subjects, and for each subject we obtained 10 facial expressions for 10 different instances of each emotion. Thus for 5 emotions, we had 50 facial expressions for individual subjects. Three out of 10 instances of emotional expression are given in Table 3 for a subject. Now, for each facial expression given in Table 3, we segmented the mouth region by Fuzzy k-means clustering algorithm, and determined the optimal lipparameters: b, c, l, p, v, n, a, h and s by adapting the model lip-contour with Table 3. Facial Expression for Subject 1 for Different Emotions Emotion Instances

Anger

Disgust

Fear

Happiness

Sadness

1.

2.

3.

Table 4. Lip Parameters for Subject 1 for Different Emotions Emotion

Instanceb

c

l

p

v

n

a

h

s

HAPPY

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

63 80 69 38 45 40 49 54 47 40 36 48 48 48 39

226 221 221 170 171 164 148 151 143 190 187 176 147 161 140

228 249 228 149 152 159 142 167 151 168 159 165 143 168 144

55 36 49 38 38 17 18 26 29 26 22 30 33 25 25

194 241 156 64 45 67 116 141 126 96 92 98 133 134 127

25 44 49 64 45 67 46 38 36 10 12 12 31 26 42

14 52 54 81 66 79 60 61 54 24 27 29 49 45 53

0 44 49 75 53 74 52 54 44 19 23 18 40 35 49

SAD

FEAR

DISGUST

ANGER

55 46 45 39 37 38 48 45 44 41 35 39 36 32 33

Emotion Recognition from the Lip-Contour of a Subject

617

the outer boundary of individual segmented lips to have an optimal matching between the two. This matching was performed by ABC algorithm. Table 4 shows the result of lip-parameters obtained from Table 3. The weight vectors for the SVM classifiers for individual emotion of a subject are then identified. This is done by first preparing a table with 10 positive and 40 negative instances for each emotion classes of a subject. The weight vector for the given SVMclassifier for the emotion class is determined in a manner, so that all the positive 2 and negative instances are separated with a margin of W  . This is done for all individual subjects separately. The second phase of the experiment starts with an unknown facial expression of a known subject. We first obtain mouth region from the image by Fuzzy K-means clustering, and determine lip-parameter by ABC using the model-lip. Now, we feed the lip-parameters to all the 5 SVM classifiers for the person concerned. The output of one or more SVM-classifiers may be +1. The decision logic then determines the emotion class of the unknown pattern.

6

Conclusion

The chapter proposed a new approach to emotion classification from the lipcontour of the subjects experiencing a specific emotion. Experiments with large number of subjects confirm that the proposed model can capture most of the experimental lip-contour for a specific emotive experience of the subject. The ABC algorithm used here is very fast and robust and thus can easily determine the parameters of the lip-contour. The SVM classifier, which is already an established tool for pattern classification with high accuracy has been utilized here for classifying lip parameters onto emotions. Experiments here too confirm that the percentage accuracy in classification of emotion on an average is 86% as obtained from the data set of 50 Indian subjects each having ten frames per emotion.

References 1. Karaboga, D., Basturk Akay, B.: Artificial Bee Colony Algorithm on Training Artificial Neural Networks. In: IEEE 15th Signal Processing and Communications Applications, pp. 1–4 (June 2007) 2. Konar, A., Chakraborty, A.: Emotional Intelligence: A Cybernetic Approach. Springer, Heidelberg (2009) 3. Bouvier, C., Coulon, P.-Y., Maldague, X.: Unsupervised Lips Segmentation Based on ROI Optimisation and Parametric Model. U.Stendhal 46 av. F. Viallet, France and LVSN, University Laval, Canada in IEEE (2007)

Software Coverage : A Testing Approach through Ant Colony Optimization Bhuvnesh Sharma, Isha Girdhar, Monika Taneja, Pooja Basia, Sangeetha Vadla, and Praveen Ranjan Srivastava Birla Institute of Technology and Science, Pilani Campus, India {bhuvnesh.bits,ishagirdhar,monicks1,pooja.basia,san3003.v, praveenrsrivastava}@gmail.com

Abstract. Software Testing is one of the most important parts of the software development lifecycle. Testing effectiveness can be achieved by the State Transition Testing (STT) and path testing which is commonly used for carrying out functional and structural testing of software systems. The tester is required to test all possible transitions and paths in the system under built. Aim of the current paper is to present an algorithm for generation of test sequences for state transitions of the system as well as path generation for CFG of the software code using the basic property and behavior of the ants. This novel approach tries to find out all the effective (or can say optimal) paths and test sequences by applying ant colony optimization (ACO) principle using some set of rules. This algorithm tries to give maximum software coverage with minimal redundancy. Keywords: Software Testing, Ant Colony Optimization (ACO), Genetic Algorithm (GA), State Transition Testing (STT), Test Data, Test Sequence, Control Flow Graph (CFG).

1

Introduction

Software Engineering [1] is an engineering discipline which focuses on developing high-quality software systems which are cost effective. It is a profession where in designing, implementation, and modifications of software are involved. Software Development Life Cycle [1] is a process of developing a system which involves different steps like investigation, analysis, design, implementation and maintenance. Through software testing an organization can gain consumers’ confidence towards the system [1]. Software testing is a labour intensive and very expensive task. It accounts almost 50 % of software development life cycle [2] [3]. Many testing tools and methodologies have been emerging in the field of software testing. Code coverage analysis is one such methodology, which helps in discovering the instructions in a software program that have been executed during a test run and helps in discovering how the testing can be further improved to cover more number of instructions during testing of a software program [2]. This paper proposes an algorithm with tool, named ESCov (Efficient Software COVerage), which uses an ACO technique [4] to generate the optimal paths, to ensure B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 618–625, 2011. © Springer-Verlag Berlin Heidelberg 2011

Software Coverage : A Testing Approach through Ant Colony Optimization

619

maximum coverage of structural as well as behavioral testing with minimal redundancy. The work is divided into two phases. In the first phase, maximum code coverage will be achieved via functional testing. For this optimized test sequences are generated by applying the ESCov, on the directed tree graph generated from the state chart diagram of the system. After that for the obtained test sequences test data is generated manually while in the second phase, the same ESCov is applied on the CFG of the software code under test. Finally, test data is applied on the generated test sequences obtained in the second phase to validate the maximum coverage.

2

Background Work

Software testing is done with many approaches such as Particle Swarm Optimization, Genetic Algorithm, and Ant Colony Optimization etc. in which ACO is one of the effective approaches. A hybrid approach using ACO and a markov software usage model [5] can produce some better results for deriving a set of independent test paths for a software system. McMinn [6] discussed application of ACO for finding sequences of transitional statements in generating test data under evolutionary testing. Another approach for state based testing using an ACO is represented [7] in which only states are considered and not complete transitions. Many AI based techniques like Neural Networks, the black-box data processing structure is very complex and it provides a slow convergence speed [8]. Although the search in genetic algorithms is not exhaustive, the algorithm fails to find a global optimum of the fitness if get stuck in local extreme [9]. Ant colony optimization is inspired by foraging behaviors of ant colonies, and target discrete optimization problem [4]. Real ants coordinate with each other by dropping and sensing a chemical level known as pheromone on the paths. Selection of path by ants depends on stochastic or probability theory. The main idea is the self organizing principle, which allows the highly coordinated behavior of real ants can be applied with the artificial agents, which collaborate to solve the complex computational problems. In the next section this paper proposed solution regarding total software coverage.

3

Proposed Algorithm

Purpose of the proposed Algorithm is providing the optimal path coverage for state transition diagram as well as CFG of the software under test. Selection of path depends upon the probability of the path. Ant will choose the path where the probability value is high. The probability value of path depends upon: probability value calculated by ant at each node for which it has pheromone value (τij), which helps other ants to make decision in the future, and heuristic information (ηij) of the path, which indicates the visibility of a path for an ant at the current vertex. In some cases if there are equal probabilities the ant will select any feasible path randomly. The architecture of the proposal is depicted in Figure 1.

620

B. Sharma et al.

Fig. 1. Architecture of ESCov

In ESCov ant will calculate the probability based on the pheromone value and heuristic value. After the selection of a particular path ant update the pheromone level as well as heuristic value. Pheromone level is increased according to last pheromone level but heuristic value is decreased according to the last heuristic value. An ant p at a vertex ‘i’ and another vertex ‘j’ which is directly connected to “i”, it means there is a path between the vertices ‘i’ and ‘j’ i.e. (ij). The use of variables and sets used in the ESCov is given below: 1. Edge set: E = {(a, b), ()…} represents the set of all the edge(s) present in the graph. 2. Vertex set: V = {a, b…} represents the set of all the nodes present in the graph. 3. Track set: T = {{a, b…}, {a, c…}…} represents the set of traversed path(s) in the graph. 4. Pheromone value: τ = τij (p) represents the pheromone level on the edge (i, j) from current vertex ‘i’ to next vertex ‘j’ for ant ‘p’. The pheromone level is updated after the particular edge traversed. This pheromone helps other ants to make decision in future. 5. Heuristic value: η = ηij (p) indicates the desirability of a path for an ant at current vertex ‘i’ to vertex ‘j’ for ant ‘p’. 6. Probability: Selection of path depends on probabilistic value of path because it is inspired by the ant behavior. Probability value of the path depends upon pheromone value τij (p) and heuristic information ηij (p) of path for ant p. There are α, β two more parameter which used to calculate the probability of a path. These parameters α and β control the desirability versus visibility. α and β are associated with pheromone and heuristic value of the paths respectively. In the ESCov, values taken for α and β is 1. An algorithm is proposed for generating optimal paths for state transition diagram and CFG as well.

Software Coverage : A Testing Approach through Ant Colony Optimization

621

Algorithm for ant p Step 1: Initialize all parameter 1.1 Set heuristic Value (η): for every edge initialize heuristic value η =1. 1.2 Set pheromone level (τ): for every edge initialize pheromone value τ =1. 1.3 α=1, β= 1 Step 2: Run the algorithm

2.3.4.4 IF outdegree > 1 AND

2.1 WHILE indegree of end node(s) is not

2.3.4.6 ELSE IF indegree = 1 AND

(no of end nodes) >1 THEN Delete the edge from E 2.3.4.5 ELSE IF indegree > 1 THEN Delete the incoming edge from E outdegree = 1 THEN

empty DO

Delete the incoming and outgoing edges

2.2 Set current = start node 2.3 LOOP Evaluation at vertex i

from E 2.3.4.7 END IF BREAK (LOOP)

2.3.1 Update the Track Set– Push the current vertex i into the track set T

2.3.5 ELSE

2.3.2 At the ith node Get next nodes

2.3.5.1 Calculate the probability of all

(check if there is a ‘self loop’) 2.3.3 IF (there is a ‘self loop’) THEN

possible edges

[τ ] × [η ] α

Pij =

2.3.3.1 Select that edge regardless of the

ij

−β

ij

α β ∑[[τ ] × [η ] ] k

−

ik

∀k ∈ feasibleset(F( p))

ik

1

probability 2.3.3.2 Move to the selected node.

2.3.5.2 Select the edge with max probability.

2.3.3.3 Enter it in track set T.

2.3.5.3 Select the node.

2.3.3.4 Delete the edge from E.

2.3.5.4

(check if an edge is there to ‘End node’.)

Update

pheromone

level

and

desirability factor of edge i,j as

2.3.4 ELSE IF (there is a direct edge to ‘end

New τij = 2 * curr pheromone level of that

node’) THEN

edge

2.3.4.1 Select that edge regardless of the

New ηij = curr heuristic value of that edge / 4

probability

2.3.5.5 SET current = selected node

2.3.4.2 Select the node

2.4 IF end node is not reached continue

2.3.4.3 Enter it in track set T

LOOP at Step 2.3 2.5 End While

4

Demonstration of Proposed Algorithm

The example work is divided into two phases: The basis of software requirement model, a state transition model constructed for a system, the algorithm is applied on directed graph under state transition diagram of the system in first phase and in second phase a CFG of the program code should be considered as input. In both the cases, an ant start from start node which would be specified by the user and it can generate a sequence of test cases. Test sequence depends upon the feasibility of path from the current vertex to other vertices and accordingly decision for further proceeding, and in the end, it give the optimal test sequences for the module under test. Here optimal means maximum coverage with minimal redundancy.

622

B. Sharma et al.

For the program taken under test, the state transition diagram and corressponding directed graph are given as below(for example) in Figure 2 and 3 respectively.

Fig. 2. The state transition diagram

Fig. 3. The directed graph for the STD

The ESCov is applied on given diagram and the test sequences obtained alongwith the values calculated at each node by the ants is given in tables (for respective ants). At node 0 (starting node) there is only one outgoing edge to node 1. Ant move to the next node since there is no need of selection. At node 1 there are three outgoing edges (1, 1) self loop, (1, 2) and (1, 3). The ant calculates the probability value using the probability formula for each edge. All edges have same probability vaue of 0.33. So ant select any edge randomly say it chooses (1,1), update the pheromone level and heuristic value and move to the node 1. Now edge (1, 1) be blocked according to the algorithm. Now at node 1 ant is left with only two options (1, 2) and (1,3). Since 2 is the end node and the algorithm makes the ant to move to the end node if it is directly connected to the current node, ant chooses (1, 2). Since the end node is encountered, the first ant stops, and the path can be calculated by traversing the edges and nodes covered by the ant. Ant 1 take the path mentioned in the table from the start node to the end node.

Similarly, the second ant generate a path from the starting node to the end node mentioned in the table

Software Coverage : A Testing Approach through Ant Colony Optimization

623

Test sequences generated by the two ants for the state transition diagram of the program under test are: Path 1: 0112

&

Path 2: 0131345

Here the ants have generated two test sequences and covered all nodes as well as edges so the coverage is 100%. Now the algorithm be applied to the CFG of the code and optimal paths would be generated. For the code under test (for example), the CFG is given in Figure 4. The path sequences obtained by applying the ESCov on the CFG are given in the tables given for individual ants. The optimal paths generated by ESCov are Path1:01234 Path2:01111213 Path 3:0123563567810 Path4:01111214 Path5: 0123563567910

Fig. 4. CFG for source code

Illustration of path table for Ant 3 is shown in tabular representation as below (for example):

624

B. Sharma et al.

Here the ants have generated five paths and covered all nodes as well as edges so the coverage is 100% with minimal redundancy since no path is repeated.

5

Analysis

The main proposed advantage of the algorithm is the maximum coverage with minimal repetition. It is tried to be achieved by avoiding self loops to be executed repeatedly reducing the redundancy. The time complexity in best case is O (n*log n), but the same in average or worst case is O (n2). There is a constraint in the ESCov in case of functional testing; it only captures single transition in between two states. If there are more than one transition in between two states, then only one is considered and rest are left untraversed.

6

Experimentation, Results and Discussion

The ESCov, to prove its essence, is compared with predefined tools (say Q) for functional coverage [11] and for structural coverage (PPTACO) [10]. Functional This paper used a class management system state machine diagram [11] as case study on which it gave 4 possible paths for transitions given in table as reported in [11]. The ESCov is then applied on the state chart diagram of the same case study which now produces 3 independent paths covering all the states and hence gives total coverage. Therefore, it shows the ESCov provides minimal number of repetitions producing maximum coverage. Structural The PPTACO tool along with path generation also calculates strength [10] for prioritizing the paths to be covered. But as shown the path generation algorithm using ACO has repetition in it but ESCov provides the same coverage with minimum repetition.

The above graph shows the difference that the number of iterations increases if there is more scope of coverage. Every iteration gives a different path in the ESCov minimizing the redundancy.

Software Coverage : A Testing Approach through Ant Colony Optimization

7

625

Conclusion

This paper proposes a model and tool named ‘ESCov’ for test sequence generation for state based testing and optimal path generation for structural testing using ant colony optimization. The result that is produced by applying proposed method is very encouraging. To model the system state chart diagram and CFG are taken and the algorithm is applied over them. After successful execution of algorithm, it shows path sequence, which gives maximum coverage and minimum redundancy. This algorithm may be very useful and helpful for the testers in the software industry. A number of extensions and applications of this model may be possible by using the different metaheuristic techniques.

References [1] [2] [3] [4] [5]

[6]

[7]

[8] [9] [10]

[11]

Sommerville, I.: Software Engineering, 8th edn. Pearson Edition (2009) Mathur, A.P.: Foundation of Software Testing, 1st edn. Pearson Education (2007) Myers, G.: The Art of Software Testing, 2nd edn., p. 234. John Wiley & Son. Inc. (2004) Dorigoa, M., Stutzle, T.: Ant colony optimization, The Knowledge Engineering Review, vol. 20, pp. 92–93. Cambridge University Press, New York (2005) Doerner, K., Gutjahr, W.J.: Extracting Test Sequences from a Markov Software Usage Model by ACO. In: Cantú-Paz, E., Foster, J.A., Deb, K., Davis, L., Roy, R., O’Reilly, U.-M., Beyer, H.-G., Kendall, G., Wilson, S.W., Harman, M., Wegener, J., Dasgupta, D., Potter, M.A., Schultz, A., Dowsland, K.A., Jonoska, N., Miller, J., Standish, R.K. (eds.) GECCO 2003. LNCS, vol. 2724, pp. 2465–2476. Springer, Heidelberg (2003) McMinn, P., Holcombe, M.: The State Problem for Evolutionary Testing. In: Cantú-Paz, E., Foster, J.A., Deb, K., Davis, L., Roy, R., O’Reilly, U.-M., Beyer, H.-G., Kendall, G., Wilson, S.W., Harman, M., Wegener, J., Dasgupta, D., Potter, M.A., Schultz, A., Dowsland, K.A., Jonoska, N., Miller, J., Standish, R.K. (eds.) GECCO 2003. LNCS, vol. 2724, pp. 2488–2500. Springer, Heidelberg (2003) Li, H., Lam, C.P.: An Ant Colony Optimization Approach to Test Sequence Generation for State based Software Testing. In: Proceedings of the Fifth International Conference on Quality Software (QSIC 2005), pp. 255–264 (2005) Briand, L.C.: On the many ways Software Engineering can benefit from Knowledge Engineering. In: Proc. 14th SEKE, Italy, pp. 3–6 (2002) Pedrycz, W., Peters, J.F.: Computational Intelligence in Software Engineering. World Scientific Publishers (1998) Srivastava, P.R., Baby, K.M.: An Approach of Optimal Path Generation using Ant Colony Optimization, pp. 1–6. IEEE-TENCON, Singapore (2009) ISBN-978-1-42444546-2 Doungsa-ard, C., Dahal, K., Hossain, A., Suwannasart, T.: An Improved Automatic Test Data Generation from UML State Machine Diagram. In: ICSEA (2007)

Short Term Load Forecasting Using Fuzzy Inference and Ant Colony Optimization Amit Jain1, Pramod Kumar Singh2, and Kumar Anurag Singh2 1

IIIT Hyderabad, Hyderabad, India 2 ABV-IIITM, Gwalior, India

Abstract. The short term load forecasting (STLF) is required for the generation scheduling and the economic load dispatch at any time. The short term load forecast calculates the power requirement pattern for the forecasting day using known, similar previous weather conditions. This paper describes a new approach for the calculation of the short term load forecast that uses fuzzy inference system which is further optimized using an Ant Colony Optimization (ACO) algorithm. It takes into account the load of the previous day, maximum temperature, average humidity and also the day type for the calculation of the load values for the next day. The Euclidean norm considering the weather variables and type of the day with weights is used to get the similar days. The effectiveness of the proposed approach is demonstrated on a typical load and weather data.

1

Introduction

Forecasting is the establishment of the future expectations by analysis of the past data, or formations of the opinions; it helps in planning and operation of the system. One of the major roles of the forecasting is in the better utilization of resources and the estimation of the benefits out of it. In general, forecasting methods may be broadly classified as (i) quantitative and (ii) qualitative. The quantitative techniques include simple regression, multiple regression, time trends, moving averages etc [1]. The qualitative methods include Delphi method, Nominal group technique, Jury of executive opinion, scenario projection etc. The rapid growth of demand for energy in the last few decades and the depletion of the fossil fuel resources have given an impetus to the development of optimal energy planning in the power system utilization. In particular, the short term load forecasting (STLF) has gained importance and has put up greater research challenges due to the recent trend of the deregulation of the electricity industry. In case of the real time dispatch operation, any forecasting error causes more electricity purchasing cost or breaking penalty cost to keep up the electricity supply and the consumption balance. Several methods for the short term load forecasting are applied like the time series, regression analysis, exponential smoothing etc [2]. The load of the power systems is dependent on the weather conditions and the social conditions. The weather conditions include temperature, humidity, chill factor and wind speed etc., whereas the social conditions include, e.g., the holiday, working hours, demographic changes and changes in the living standards. Heinemann et al. [3] B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 626–636, 2011. © Springer-Verlag Berlin Heidelberg 2011

Short Term Load Forecasting Using Fuzzy Inference and Ant Colony Optimization

627

have shown that the ratio of the summer to winter system peak load increased from 90% in the year 1953 to 111% in the year 1964. The daily peak load can be mainly divided into two types of load that are basic load and the weather sensitive load [4]. The system load has been responsive to changes in the weather conditions; hence the system operators must be prepared for summer afternoons having the sudden large increase in the demands. A large percentage of summer weather sensitive loads depend on the relative degree of comfort or discomfort of the human beings. The treatment of winter daily system peak loads would be very similar in details [5]. To make such forecasts, it is necessary to examine the history of each component to establish the trends. The above factors complicate the load forecasting. We observe that, in the literature, multiple linear regression analysis based on the value of the previous load has traditionally been used for load forecasting. Recently, methods based on artificial intelligence, e.g., artificial neural networks, fuzzy logic, have also been discussed in the literature [6], [7], [8], [9], [10]. The artificial neural network does to produce promising result for STLF as it fails to respond properly with respect to rapid fluctuations in the data. Though there are many weather variables which affect the load, the major deciding variables are the temperature and the humidity [11]. In this paper, the proposed model takes into account the past month’s temperature and humidity, load of the previous hour, temperature of the present hour, temperature of the previous hour, average humidity of the present as well as the previous hour and the similar day types. This paper is organized as follows. Section 2 includes the data analysis part. The proposed forecasting method is presented in Section 3. Section 4 presents simulation results and is followed by the conclusion in Section 5.

2

Variables Affecting the Load Pattern

The analysis is carried out on data containing hourly values of the load, and average temperature and average humidity during that hour. A relationship between the load and the weather variables is presented here. 2.1

Variation of Load with Temperature

One of the most important weather variables that influence the load is temperature. A scatter plot of the given load and the hourly temperature values is presented in Fig. 1. It shows a positive correlation between the load and the temperature, i.e., demand of load increases as the temperature increases. 2.2

Variation of Load with Humidity

Humidity is another important weather variable which affects the load. Fig. 2 represents a scatter plot of the average load and the average humidity. Here too, it is observed that the correlation between the load and the humidity is positive, i.e., the demand of load increases as the humidity increases.

628

A. Jain, P.K. Singh, and K.A. Singh

Fig. 1. Plot between average load Vs average temperature

Fig. 2. Plot between the average load Vs average humidity

2.3

Relation between the Present and the Previous Hour Load

The present hour load is dependent on the previous hour load as well as the load of the same hour of the previous day. Hence, in multiple linear regression models the effect of the previous hour load has also been considered. It is assumed that there is no significant change in the seasonal weather in one day and also not significant change in temperature or humidity in one hour. Accordingly, the pattern of load consumption will also be similar to the previous load consumptions. Fig. 3 represents variation of the load consumption with the hours. It shows a cyclic behavior of the load consumption pattern; load consumption in weekdays is different from that of the weekend.

Short Term Load Forecasting Using Fuzzy Inference and Ant Colony Optimization

629

Fig. 3. The load consumption pattern Vs the hours

3

Short Term Load Forecasting Using Fuzzy Inference Followed by Ant Colony Optimization

3.1

Similar Day Selection

To calculate the similarity between the forecast day and the searched previous days, the Euclidean norm with the weight factors is used. The value of the Euclidean norm is inversely proportional to the similarity, i.e., lesser is the value of the Euclidean norm more similar is that particular day to the forecast day. The Euclidean norm uses the maximum temperature, average humidity and the day type, each of which is associated with the weights. It gives a better method of the selection of the similar days as it contains the temperature, the humidity and also the day type. The weight factors associated with the temperature, humidity and the day type are calculated based on the historic data by using regression analysis on it. The expression for the Euclidean norm is given by: EN = (w1 * (ΔT) 2 + w2 * (ΔH) 2 + w3 * (ΔD) 2)0.5

(1)

ΔTmax = Tmax -

Tmaxp

(2)

ΔHavg = Havg –

Havgp

(3)

ΔD = D - DP

(4)

Here, Tmax and Havg are maximum temperature and average humidity respectively for the forecast day, Tmaxp and Havgp are maximum temperature and average humidity of the searched previous days. The weight factors w1, w2 and w3 are evaluated using the regression analysis on the historical data. The weights are used in the formula of the Euclidean norm for the scaling purpose.

630

3.2

A. Jain, P.K. Singh, and K.A. Singh

Fuzzy Inference System

The Euclidean norm alone should not be used for the forecasting process as it can have large value of mean absolute percentage error. Assuming the same trends of the relationship between the forecast day and the previous similar days as that of the current day forecast and its similar days, the similar days can be evaluated by using the previous forecast day and its previous similar days. The fuzzy inference system is used to evaluate the similarity between the previous forecast day and the previous similar days which results in the correction factors. The correction factors are used to correct the similar days of the forecast day to obtain the value of the load forecast. To evaluate the degree of the similarity, 3 fuzzy input variables are taken namely: ELk = LP – Lpsk

(5)

ETk = Tp – Tpsk

(6)

k

EH = Hp –

Hpsk

(7)

Here, Lp and Lps are the average load of the previous forecast and the previous kth similar day respectively. Similarly, Tp, Tps, Hp, and Hps are the corresponding variables for the temperature and humidity respectively. EL, ET and EH take the fuzzy set values low, medium and high. The fuzzy rules for the fuzzy inference system are based on the generalized knowledge of the effect of each of the variable on the load. If the membership of EL is µ EL that of ET is µ ET and EH is µ EH then the firing strength µ can be calculated using the min operator. i

i

i

µ = min (µEL , µET , µEH )

(8)

The crisp value of the correction factors are then obtained by the centroid defuzzyfication scheme Wk = ∑ αi µ ik / ∑ µ ik

(9) th

Wk is the correction factor for the load curve on the k similar day to shape the forecast day and the forecast curve for the next day is given by k

L (t) = 1/N [∑ (1 + Wk) * Ls (t)]

(10)

Here, N is the number of the similar days taken and ‘t’ is hourly time from 1 to 24. The forecast results deviation from the actual values are represented in the form of Mean absolute percentage error (MAPE) [5], [6]. It is defined as follows: MAPE = (1/N) * (∑ ( |PA - PF| / PA )) *100

(11)

Here, PA and PF are the actual and the forecast values of the load, and N is the number of hours of the day i.e. N = 1, 2, 3… 24. 3.3

Ant Colony Optimization

Ant colony optimization (ACO) is a paradigm for designing meta heuristic algorithms for combinatorial optimization problems. The main underlying idea, loosely inspired by the behavior of the real ants is that of a parallel search over several constructive

Short Term Load Forecasting Using Fuzzy Inference and Ant Colony Optimization

631

computational threads based on local problem data and on a dynamic memory structure containing information on the quality of the previously obtained result. A combinatorial optimization problem is a problem defined over a set C = c1, c2, …, cn of the basic components. A subset S of the component represent the solution of the problem. F Subset of 2c is a subset of feasible solutions, thus a solution S is feasible if and only if S belongs to F. A cost function Z is defined over the solution domain Z : 2c -> R, the objective being to find a minimum cost feasible solution S* i.e. to find S* : S* belongs to F and Z(S*) 0 ∧ ∀ y (1 −

d (oi , o j )

α

)>0

(4)

otherwise

f ( xi ) > 0 ∧ ∀ y (1 −

|| x − y ||

α

) > 0 has been introduced to

eliminate the possibility of dropping the object in the vicinity of δ elements significantly different from it. The value of increasing the probability of raising an object is described here by the equation:

1, when f ( xi ) ≤ 1  Pp =  1  f ( x ) 2 , otherwise i 

(5)

and the probability of dropping an object is described as:

1, when f ( xi ) ≥ 1, Pd =  4  f ( xi ) , otherwise

(6)

This approach ensures that the operations of raising and dropping objects under consideration are deterministic, for a very little value of the density function in the first case and a very high value in the second case. It is connected with the process accelerating the formation of clusters, but only in areas which consist of a large number of objects with similar attribute vectors. Moreover, in the algorithm, the process of clustering has been further accelerated by the introduction of memory for each ant enabling it to store information about previously performed actions, i.e., about the modification of positions of objects being transported earlier.

2

The Approach

Despite many significant advantages such as no need to pre-determine the number of clusters, the algorithm described here has also several disadvantages. The main one is a lack of stability of the obtained solutions. This is reflected, especially when we examine the data space with a small cardinality. Moreover, in case of a large number of collections, but with their low frequencies, the use of this algorithm leads to a number of resulting groups smaller than the real one. Can we fix this? The suggestion

640

A. Lewicki, K. Pancerz, and R. Tadeusiewicz

to solve this problem, which was proposed by the authors of the publication and preceded by empirical studies, involves modifying the linear dependence of the correlation of the main parameters of the algorithm, which are: a function of dissimilarity of compared objects and the α parameter indirectly associated with it, and the ratio of the tested neighborhood. A correlation value of the parameters for m elements of the considered data set can be defined as: m

 (x

r=

i =1

m

 (x

i

i

− x )( y i − y )

− x)

(7)

2 m

i =1

(y

i

− y)2

i =1

which leads to the simplified form:

r=

where

1 m  xi y i − x y n i =1 1 m 2 1 m 2 (  xi − x 2 ) (  yi − y 2 ) n i =1 n i=1

(8)

x, y are appropriate average values defined as: x=

1 m 1 m , x y =  i  yi n i =1 n i =1

(9)

In the ant-based clustering algorithm proposed previously by the authors, a similarity measure of feature vectors of compared objects takes into account the Euclidean metric. Meanwhile, the disadvantage of this approach is not only omitting vector components with very small dimensions, but also a very large computational time. Therefore, considering the components represented by the feature vectors of the analyzed elements, there can be applied here the previously unexplored approach based on metrics associated with the Jacquard coefficient, where the similarity of objects oi and oj is represented by the corresponding vectors of features.

 +

m

d (oi , o j ) = 1 −



m

x k =1 ik

2

x * x jk

k =1 ik m 2 k =1 jk

x

− k =1 xik x jk m

(10)

where xik and xjk represent the k-th feature of the vectors of objects oi and oj, respectively. In view of the fact that, in the modified algorithm, the degree of dissimilarity of objects is taken into account, the formula must be converted to the form:

1 k =1 xik * x jk ) d ( oi , o j ) = 1 − ( m 2  xik 2 + m x jk 2 − m xik x jk k =1 k =1 k =1 m

(11)

The Use of Strategies of Normalized Correlation

641

Another proposed change concerns the factor that defines the size of the tested neighborhood of the available solutions. Observation made previously, demonstrated the fact that, with increasing radius of ant perception, the number of objects, for which the value of the dissimilarity function is calculated (11), also increases. Therefore, this factor may be designated as d2 *δ2 for d which is the quotient of the smallest and the largest distance to the initialization size of the neighborhood. The formula defining the density function has the form:

d (oi , o j ) d (oi , o j )  2 ), when f ( xi ) > 0 ∧ ∀ y (1 − )>0 d * δ  (1 − (12) α α f ( xi ) =  y∈N χ ( i )  otherwise 0, The introduced modification should decrease the importance of the distance between the considered objects with increasing the radius of perception and, dependent on it, the size of the area of studied solutions. Evaluation tests of the proposed solution were carried out on three data sets (labeled with S1, S2 and S3). They contained 900, 400 and 90 objects, respectively, characterizing feature vectors which are attributes representing economic data on financial markets shared by WBK. Optimal values of the main parameters of the algorithm determined by manual tuning on the sample data model are presented in Table 1. Table 1. The adopted values of the initialization parameters of the implemented algorithm

Parameters Number of ants Memory size Step size α δ

Value 200 30 100 0.73 2

In order to evaluate the results, the authors of the article compared the proposed method with the original ATTA algorithm and the reference k-means algorithm. The quality of the process of aggregation was measured using parameters such as the Dunn index, rand and F-measure.

3

Results

The results of the assessment of the proposed algorithm, the ATTA algorithm and the k-means algorithm made with the usage of three indexes for examined data sets are presented in Table 2. The results show clearly that the algorithm works well for smaller data sets, while for high-cardinality data the choice of k-means algorithm would be a qualitatively better approach. Nevertheless, the evaluation process demonstrated an advantage of the proposed modifications over the standard ATTA algorithm for each of the three test sets.

642

A. Lewicki, K. Pancerz, and R. Tadeusiewicz

Table 2. Evaluation of the quality of the proposed algorithm compared to the ATTA algorithm and the k-means algorithm

Data set S1 S1 S1 S2 S2 S2 S3 S3 S3

Index Dunn index Rand F-measure Dunn index Rand F-measure Dunn index Rand F-measure

ATTA algorithm 1.031 0.563 0.676 0.752 0.504 0.473 1.227 0.886 0.764

K-means algorithm 1.211 0.578 0.705 0.983 0.560 0.678 0.770 0.875 0.747

The proposed algorithm 1.116 0.576 0.706 0.708 0.522 0.574 1.391 0.889 0.774

In order to determine the impact of the target strategy and the approach based on the standard ant-based clustering algorithm to the linear dependency of parameters of the algorithm, there were also made calculations of the value of the correlation coefficient formula described previously (7). These results are presented in Table 3. Table 3. The impact of the proposed solution to the linear correlation of the main parameters of the algorithm

Algorithm ATTA ATTA ATTA The proposed algorithm The proposed algorithm The proposed algorithm

Data set S1 S2 S3 S1 S2 S3

The correlation coefficient 0.632280 0.415771 0.486394 0.132473 0.106767 0.124522

An analysis of the results listed in Table 3 shows that the proposed approach resulted also in a significant reduction of the coefficient of linear correlation between the parameters of the algorithm, which, for the standard ATTA algorithm, had a noticeable impact because the value (which is up to 1.0) oscillated here within 0.41 to 0.63.

4

Conclusions

Modifications of the algorithm proposed by the authors, inspired by the behavior of ants, whose main advantage is the return of satisfactory results, despite the lack of information about the number of resulting classes, show that a solution based on the metric with the Jacquard factor and modification of a decisive factor for the vicinity of the tested space have influence on reducing the linear dependence of the main parameters of the algorithm. For the studied data sets, a solution found by the algorithm was also burdened with an error smaller than for the original algorithm and

The Use of Strategies of Normalized Correlation

643

the k-means algorithm. Although experiments have shown that the k-means algorithm achieved significantly better results than the accepted method of solving a defined problem, it should be noted that, in case of a non-deterministic algorithm, we cannot talk about the knowledge of the available data structures, while its advantages can include independence from the dimensionality of data. Therefore, in the next step in the near future, research should attempt to determine the quality of the proposed solutions for incomplete data sets and data sets containing clusters of an irregular shape. For this type of data, there can be proved that, in fact, the division into the appropriate clusters is not strongly dependent on the modified parameters of the algorithm, and a very important role may be played by a selection of fine-tuning strategies for the α coefficient. It seems to be probable that the case, when features describing the objects being compared are characterized by a different factor of importance, may be problematic. Therefore, further experiments should include the use of a metric taking into account the importance of a given attribute of the object under consideration, with the possibility of using measures of similarity for the features of various types. Acknowledgements. This paper has been partially supported by the grant No. N N519 654540 from the National Science Centre in Poland.

References 1. Hore, P.: Distributed clustering for scaling classic algorithms, Theses and Dissertations, University of South Florida (2004) 2. Lewicki, A., Tadeusiewicz, R.: The recruitment and selection of staff problem with an Ant Colony System, Backgrounds and Applications. AISC, vol. 2. Springer, Heidelberg (2010) 3. Lewicki, A., Generalized non-extensive thermodynamics to the Ant Colony System, Information Systems Architecture and Technology, System Analysis Approach to the Design, Control and Decision Support, Wroclaw (2010) 4. Lewicki, A.: Non-Euclidean metric in multi-objective Ant Colony Optimization Algorithms, Information Systems Architecture and Technology, System Analysis Approach to the Design, Control and Decision Support, Wroclaw (2010) 5. Lewicki, A., Tadeusiewicz, R.: An autocatalytic emergence swarm algorithm in the decision-making task of managing the process of creation of intellectual capital. Springer, Heidelberg (2011) 6. Handl, J., Knowles, J., Dorigo, M.: Ant-based clustering and topographic mapping. Artif. Life 12(1) (2006) 7. Decastro, L., Von Zuben, F.: Recent Developments In Biologically Inspired Computing. Idea Group Publishing, Hershey (2004) 8. Mohamed, O., Sivakumar, R.: Ant-based Clustering Algorithms: A Brief Survey. International Journal of Computer Theory and Engineering 2(5) (October 2010) 9. Dorigo, M., Di Caro, G., Gambarella, L.: Ant Algorithms for Discrete Optimization. Artificial Life 5(3) (1999) 10. Azzag, H., Monmarché, N., Slimane, M., Venturini, G., Guinot, C.: AntTree: A new model for clustering with artificial ants. In: IEEE Congress on Evolutionary Computation, vol. 4, pp. 2642–2647. IEEE Press, Canberra (2003)

644

A. Lewicki, K. Pancerz, and R. Tadeusiewicz

11. Scholes, S., Wilson, M., Sendova-Franks, A., Melhuish, C.: Comparisons in evolution and engineering: The collective intelligence of sorting. Adaptive Behavior - Animals, Animats, Software Agents, Robots, Adaptive Systems 12(3-4) (2004) 12. Sendova-Franks, A.: Brood sorting by ants: two phases and differential diffusion. Animal Behaviour (2004) 13. Boryczka, B.: Ant Clustering Algorithm, Intelligent Information Systems. Kluwer Academic Publishers (2008) 14. Abbass, H., Hoai, N., McKay, R.: AntTAG, A new method to compose computer using colonies of ants. In: Proceedings of the IEEE Congress on Evolutianory Computation, Honolulu, vol. 2 (2002) 15. Vizine, A., de Castro, L., Hruschka, E., Gudwin, R.: Towards improving clustering ants: An adaptive clustering algorithm. Informatica Journal 29 (2005) 16. Ouadfel, S., Batouche, M.: An Efficient Ant Algorithm for Swarm-based Image Clustering. Journal of Computer Science 3(3) 17. Deneubourg, J., Goss, S., Franks, N., Sendova-Franks, A., Detrain, C., Chretien, L.: The dynamics of collective sorting robot-like ants and ant-like robots. In: Proceedings of the First International Conference on Simulation of Adaptive Behavior: From Animals to Animats. MIT Press, Cambridge (1990) 18. Das, S., Abraham, A., Konar, A.: Metaheuristic Clustering. Springer, Heidelberg (2009)

Ant Based Clustering of Time Series Discrete Data – A Rough Set Approach Krzysztof Pancerz1, Arkadiusz Lewicki1 , and Ryszard Tadeusiewicz2 1

University of Information Technology and Management in Rzesz´ ow, Poland {kpancerz,alewicki}@wsiz.rzeszow.pl 2 AGH University of Science and Technology, Krak´ ow, Poland [email protected]

Abstract. This paper focuses on clustering of time series discrete data. In time series data, each instance represents a different time step and the attributes give values associated with that time. In the presented approach, we consider discrete data, i.e., the set of values appearing in a time series is finite. For ant-based clustering, we use the algorithm based on the versions proposed by J. Deneubourg, E. Lumer and B. Faieta. As a similarity measure, the so-called consistency measure defined in terms of multistage decision transition systems is proposed. A decision on raising or dropping a given episode by the ant is made on the basis of a degree of consistency of that episode with the knowledge extracted from the neighboring episodes. Keywords: ant based clustering, consistency measure, episodes, rough sets, time series.

1

Introduction

In data mining, we encounter a diversity of methods supporting data analysis which can be generally classified into two groups, called supervised and unsupervised learning, respectively (cf. [2]). One of unsupervised approaches investigated by us is clustering of data. In the paper, we consider the ant-based clustering algorithm based mainly on versions proposed by Deneubourg [4], Lumer and Faieta [9]. The ant-based clustering algorithm discovers automatically clusters in numerical data without prior knowledge of possible number of clusters. It is a very important feature in the preliminary step of data analysis. Clustering algorithms examine data to find groups (clusters) of items (objects, cases) which are similar to each other and dissimilar to the items belonging to other groups. The concept of dissimilarity or dual similarity is the essential component of any form of clustering that helps us to navigate through the data space and form clusters. We try to incorporate into the ant-based clustering algorithm a consistency measure defined for temporal data (see [12], [13]). This consistency measure plays a role of a similarity measure. The consistency factor is defined in terms of multistage decision transition systems (MDTSs) proposed in [12] to describe transitions among states observed B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 645–653, 2011. c Springer-Verlag Berlin Heidelberg 2011 

646

K. Pancerz, A. Lewicki, and R. Tadeusiewicz

in the given systems. If we are interested in sequences of changes of states, then we may represent such changes by means of polyadic transition relations over the sets of states. A given MDTS represents such a relation. Each object in MDTS is referred to as an episode. We can extend a given MDTS by adding new episodes to it. Next, we can answer an important question: ”what is a consistency factor of a new episode added to MDTS with the knowledge included in MDTS?”. A proper algorithm enabling us to answer this question is shown in Section 3. We consider time series discrete data as the data to be clustered. In time series data, each instance represents a different time step and the attributes give values associated with that time. In the presented approach, we consider discrete data, i.e., the set of values appearing in a time series is finite. It follows from the approach to calculating the consistency measure. We assume that multistage decision transition systems include only discrete values of attributes. In case of continuous values, there is a need to define another relation between episodes. In the literature, there have been proposed various algorithms combining clustering with fuzzy or rough set theory (cf. [3], [10], [11]). Our method differs from those proposed in the literature as we use a different way to compute a similarity measure (in our approach, it is based on the knowledge expressed by the rules determining transitions between states in time series data).

2

Basic Notions

A concept of an information system is one of the basic concepts of rough set theory. Information systems are used to represent some knowledge of elements of a universe of discourse. An information system is a pair S = (U, A), where U is a set of objects, A is a set of attributes, i.e., a : U → Va for a ∈ A, where Va is called a value set of a. A decision system is a pair S = (U, A), where A = C ∪ D, C ∩ D = ∅, and C is a set of condition attributes, D is a set of decision attributes. Any information (decision) system can be represented as a data table whose columns are labeled with attributes, rows are labeled with objects, and entries of the table are attribute values. Let S = (U, A) be an information system. Each subset B ⊆ A of attributes determines an equivalence relation on U , called an indiscernibility relation Ind(B), defined as Ind(B) = {(u, v) ∈ U × U : ∀ a(u) = a(v)}. The equivalence class a∈B

containing u ∈ U is denoted by [u]B . Let X ⊆ U and B ⊆ A. The B-lower approximation BX of X and the Bupper approximation BX of X are defined as BX = {u ∈ U : [u]B ⊆ X} and BX = {u ∈ U : [u]B ∩ X = ∅}, respectively. A temporal information system is a kind of an information system S = (U, A), with a set U of objects ordered in time, i.e., U = {ut : t = 1, 2, . . . , n}, where ut is the object observed in time t. By a time window on S of the length λ in a point τ we understand an information system S  = (U  , A ), where U  = {uτ , uτ +1 , . . . , uτ +λ−1 }, 1 ≤ τ , τ + λ − 1 ≤ n, and A is a set of all attributes from A defined on the domain restricted to U  . The length λ of S  is defined as λ = card(U  ). In the sequel, the set A of all attributes in any time window

Ant Based Clustering of Time Series Discrete Data – A Rough Set Approach

647

S  = (U  , A ) on S = (U, A) will be marked, for simplicity, with the same letter A like in S.

3

Consistency Measure

A dynamic information system proposed by Z. Suraj in [15] includes information about transitions between states observed in a given system. It means that the dynamics is expressed by a transition relation defined in a dynamic information system and the term of a dynamic information system should be understood in this sense. Here, we give some crucial notions concerning dynamic information systems. A multistage transition system is a pair M T S = (U, T ), where U is a nonempty set of states and T ⊆ U k is a polyadic transition relation, where k > 2. A multistage dynamic information system is a tuple M DIS = (U, A, T ), where S = (U, A) is an information system called the underlying system of M DIS and M T S = (U, T ) is a multistage transition system. The underlying system includes states observed in a given system whereas a transition system describes transitions between these states. Let S = (U, {a}) be a temporal information system representing a time series, where U = {ut : t = 1, 2, . . . , n}. On the basis of S, we can define a multistage transition system M T S, and next, a multistage dynamic information system M DIS. S is an underlying system for M DIS. For each τ = 1 . . . n − λ + 1, we create a time window S  = (U  , {a}), where U  = {uτ , uτ +1 , . . . , uτ +λ−1 }. We obtain n−λ+1 time windows of the length λ. All sequences (uτ , uτ +1 , . . . , uτ +λ−1 ) of objects define a polyadic transition relation T ⊆ U λ . Each element of a multistage transition relation T in a multistage dynamic information system M DIS = (U, A, T ) is a sequence of states (from the set U ), which can be referred to as an episode. The episode is, in fact, a fragment of a time series. This fragment has a fixed length. Let M DIS = (U, A, T ) be a multistage dynamic information system, where T ⊆ U k . Each element (u1 , u2 , . . . , uk ) ∈ T , where u1 , u2 , . . . , uk ∈ U , is called an episode in M DIS. We can use a suitable data table to represent a multistage transition system. Such a table will represent the so-called multistage decision transition system. Let M T S = (U, T ) be a multistage transition system. A multistage decision transition system is a pair M DT S = (UT , A1 ∪ A2 ∪ . . . ∪ Ak ), where each t ∈ UT corresponds exactly to one element of the polyadic transition relation T whereas attributes from the set Ai determine values from the i-th domain of T , where i = 1, 2, . . . , k. If we consider a time series described by one attribute, then each episode t ∈ UT in M DT S = (UT , A1 ∪ A2 ∪ . . . ∪ Ak ), where A1 = {a1 },  A2 = {a2 }, ..., Ak = {ak }, can be presented as a vector a1 (t), a2 (t), . . . , ak (t) . For a given multistage decision transition system, we can consider its elementary decision transition subsystems defined as follows. An elementary decision transition subsystem of a multistage decision transition system M DT S = (UT , A1 ∪ A2 ∪ . . . ∪ Ak ) is a decision transition system DT S(i, i + 1) = (UT , Ai ∪ Ai+1 ), where: i ∈ {1, ..., k − 1}.

648

K. Pancerz, A. Lewicki, and R. Tadeusiewicz

Now, we recall some important notions concerning extensions of multistage decision transition systems given in [12]. Any nontrivial extension of a given multistage decision transition system M DT S = (UT , A1 ∪ A2 ∪ . . . ∪ Ak ) includes new episodes such that for each episode t∗ we have a(t∗ ) ∈ Va for each a ∈ (A1 ∪ A2 ∪ . . . ∪ Ak ). Let DT S(i, i + 1) = (UT , Ai ∪ Ai+1 ) be the elementary decision transition subsystem. For each attribute a ∈ Ai and the new episode t∗ , we can transform DT S(i, i + 1) into the system with irrelevant values of attributes. If a(t∗ ) = a(t), where t ∈ UT , then we replace a(t) by the value * (denoting an irrelevant value). This means that we create a new system for which appropriate sets of attribute values are extended by the value *. The transformed system can be treated as an incomplete system. Therefore, instead of an indiscernibility relation and equivalence classes, we use a characteristic relation and characteristic sets (cf. [6]). For the transformed elementary decision transition subsystem DT S(i, i + 1) = (UT , Ai ∪ Ai+1 ), we define a characteristic relation R(Ai ). R(Ai ) is a binary relation on UT defined as R(Ai ) = {(t, v) ∈ UT2 : [∃a∈Ai a(t) = ∗] ∧ [∀a∈Ai (a(t) = ∗ ∨ a(t) = a(v))]}. For each t ∈ UT , a characteristic set KAi (t) has the form KAi (t) = {v ∈ UT : (t, v) ∈ R(Ai )}. For any X ⊆ UT , the Ai -lower approximation of X is determined as Ai X = {t ∈ UT : KAi (t) = ∅ ∧ KAi (t) ⊆ X}. Let DT S(i, i + 1) = (UT , Ai ∪ Ai+1 ) be an elementary decision transition subsystem, a ∈ Ai+1 , and va ∈ Va . By Xava we denote the subset of UT such that Xava = {t ∈ UT : a(t) = va }. For each episode t∗ from the extension M DT S, we define a consistency factor of t∗ (see [12], [13]). For a given multistage decision transition system M DT S = (UT , A1 ∪A2 ∪. . .∪Ak ) we create a family DTS of elementary decision transition subsystems, i.e., DTS = {DT S(i, i + 1) = (UT , Ai ∪ Ai+1 )}i=1,...,k−1 . Next, the consistency factor ξDT S(i,i+1) (t∗ ) of the episode t∗ with the knowledge included in DT S(i, i + 1) is computed for each subsystem DT S(i, i + 1) from the family DTS. Finally, the consistency factor ξMDT S (t∗ ) of the episode t∗ with the knowledge included in M DT S is calculated as: k−1

ξMDT S (t∗ ) = O ξDT S(i,i+1) (t∗ ), i=1

where O denotes an aggregation operator. The consistency factor ξDT S(i,i+1) (t∗ ) of the episode t∗ with the knowledge included in DT S(i, i + 1) is calculated as: ξDT S(i,i+1) (t∗ ) = 1 − where:

T = U





a∈Ai+1

va ∈Va

T ) card(U , card(UT )

{Ai Xava : Ai Xava = ∅ ∧ a(t∗ ) = va }.

A consistency factor may be calculated using Algorithm 1. In this algorithm, we assume an arithmetic average as an aggregation operator.

Ant Based Clustering of Time Series Discrete Data – A Rough Set Approach

649

Algorithm 1. Algorithm for computing a consistency factor of an episode belonging to the extension of a given multistage decision transition system Input : A multistage decision transition system M DT S = (UT , A1 ∪ A2 ∪ . . . ∪ Ak ), an episode t∗ from any extension of M DT S. Output: A consistency factor ξM DT S (t∗ ) of the episode t∗ with the knowledge included in M DT S. s ←− 0; for each elementary decision transition subsystem DT S(i, i + 1) = (UT , Ai ∪ Ai+1 ) of M DT S do Compute ξDT S(i,i+1) (t∗ ) using Algorithm 3 (treat DT S(i, i + 1) as a decision system and t∗ as an object added to the decision system); s ←− s + ξDT S(i,i+1) (t∗ ); end s ξM DT S (t∗ ) ←− k−1

Algorithm 2. Algorithm for computing a consistency factor of an object belonging to the extension of a decision system Input : A decision system S = (U, A ∪ D), a new object u∗ added to S. Output: A consistency factor ξS (u∗ ) of the object u∗ with the knowledge included in S.  ←− ∅; U for each d ∈ D do Create a copy S  = (U, A ∪ D) of S; for each u ∈ U in S  do for each a ∈ A in S  do if a(u) = a(u∗ ) then a(u) ←− ∗; end end end Remove each object u ∈ U in S  such that ∀ a(u) = ∗; for each vd ∈ Vd do Xdvd ←− {u ∈ U : d(u) = vd }; if CXdvd = ∅ then if d(u∗ ) = vd then  ←− U  ∪ CX vd ; U d end end end end ξS (u∗ ) ←− 1 −

 card(U) ; card(U )

a∈A

650

K. Pancerz, A. Lewicki, and R. Tadeusiewicz

Algorithm 3. Algorithm for Ant Based Clustering for each episode t ∈ UT do Place t randomly on a grid G; end for each ant aj ∈ Ants do Place aj randomly on a grid place occupied by one of episodes from UT ; end for k ← 1 to nit do for each ant aj ∈ Ants do if aj is unladen then if place of aj is occupied by dropped episode t then Draw a random real number r ∈ [0, 1]; if r ≤ ppick (t) then set t as picked; set aj as carrying the episode; end else move aj randomly to another place occupied by one of episodes from UT ; end else Draw a random real number r ∈ [0, 1]; if r ≤ pdrop(t) then move t randomly to a new place on a grid; set t as dropped; set aj as unladen; end end end end

4

Ant Based Clustering

The ant based clustering algorithm used in our experiment is mainly based on the versions proposed by Deneubourg [4], Lumer and Faieta [9]. There has been introduced a number of slight modifications that adjust the algorithm to the considered problem. For more details, the reader is referred to papers devoted to different studies on features and modifications of Lumer and Faieta’s algorithm for data clustering (for example, see [1], [5], [7], [8]). Here, we present only the most significant information. Algorithm 3 may be used to cluster episodes appearing in a given time series. Let M DT S = (UT , A1 ∪ A2 ∪ . . . ∪ Ak ) be a multistage decision transition system representing episodes in a time series. UT is a set of episodes being clustered (the size of UT is n), nit is a number of iterations performed for the clustering process, Ants is a set of ants used in the clustering process, ppick (t) and pdrop (t) are probabilities for the picking and dropping operations made for a given episode t, respectively (see Formulas 1 and 2).

Ant Based Clustering of Time Series Discrete Data – A Rough Set Approach

651

For picking and dropping decisions the following threshold formulas are used  1−floc (t) if floc (t) < 0.95 ppick (t) = 1+floc (t) (1) 0.95 otherwise and 4 pdrop (t) = floc (t)

(2)

respectively, where floc is a neighborhood function. floc (t) = ξMDT S ∗ (t), where M DT S ∗ includes only those episodes from M DT S which are placed in a local neighborhood of t. This local neighborhood is built from the cells on a grid surrounding locally the cell occupied by the episode t.

5

Experiments

Basic experiments validating the proposed approach were carried out on artificial data. We have clustered different time series (episodes) of three shapes: sinusoidal, cosinusoidal and exponential. Time series have been transformed into the so-called delta representation, i.e., attribute values have been replaced with differences between current values and values of previous instances. After transformation, each time series was a sequence consisting of three values: -1 (denoting decreasing), 0 (denoting a lack of change), 1 (denoting increasing). In this case, we have obtained multistage decision transition systems with discrete values. Episodes have been clustered on two-dimensional grid of size 50. An exemplary result of clustering is shown in Figure 1. Crosses correspond to sinusoidal episodes, diamonds correspond to exponential episodes, and circles correspond to cosinusoidal episodes.

Fig. 1. An exemplary result of clustering

652

6

K. Pancerz, A. Lewicki, and R. Tadeusiewicz

Conclusions

In the paper, we have examined a problem of clustering time series discrete data using the ant based approach. As a similarity measure, a consistency measure, defined in terms of multistage decision transition systems, has been used. In the future, we will apply the proposed approach to real-life data. Moreover, it seems to be necessary to tune a consistency measure to the considered problem. Acknowledgments. This paper has been partially supported by the grant No. N N519 654540 from the National Science Centre in Poland.

References 1. Boryczka, U.: Finding groups in data: Cluster analysis with ants. Applied Soft Computing 9(1), 61–70 (2009) 2. Cios, K., Pedrycz, W., Swiniarski, R.W., Kurgan, L.: Data mining. A knowledge discovery approach. Springer, New York (2007) 3. Das, S., Abraham, A., Konar, A.: Metaheuristic Clustering. Springer, Heidelberg (2009) 4. Deneubourg, J., Goss, S., Franks, N., Sendova-Franks, A., Detrain, C., Chr´etien, L.: The dynamics of collective sorting: Robot-like ants and ant-like robots. In: Proceedings of the First International Conference on Simulation of Adaptive Behaviour: From Animals to Animats, vol. 1, pp. 356–365. MIT Press, Cambridge (1991) 5. Gilner, B.: A comparative study of ant clustering algorithms (2007) 6. Grzymala-Busse, J.W.: Data with Missing Attribute Values: Generalization of Indiscernibility Relation and Rule Induction. In: Peters, J.F., Skowron, A., Grzymala´ Busse, J.W., Kostek, B.z., Swiniarski, R.W., Szczuka, M.S. (eds.) Transactions on Rough Sets I. LNCS, vol. 3100, pp. 78–95. Springer, Heidelberg (2004) 7. Handl, J., Knowles, J., Dorigo, M.: Ant-based clustering and topographic mapping. Artificial Life 12(1), 35–62 (2006) 8. Handl, J., Meyer, B.: Ant-based and swarm-based clustering. Swarm Intelligence 1, 95–113 (2007) 9. Lumer, E., Faieta, B.: Diversity and adaptation in populations of clustering ants. In: Proceedings of the Third International Conference on Simulation of Adaptive Behaviour: From Animals to Animats, vol. 3, pp. 501–508. MIT Press, Cambridge (1994) 10. Mitra, S., Banka, H., Pedrycz, W.: Rough-fuzzy collaborative clustering. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics 36, 795–805 (2006) 11. Mitra, S., Pedrycz, W., Barman, B.: Shadowed c-means: Integrating fuzzy and rough clustering. Pattern Recognition 43, 1282–1291 (2010) 12. Pancerz, K.: Extensions of dynamic information systems in state prediction problems: the first study. In: Magdalena, L., Ojeda-Aciego, M., Verdegay, L. (eds.) Proceedings of the 12th International Conference on Information Processing and Management of Uncertainty in Knowledge-based Systems (IPMU 2008), Malaga, Spain, pp. 101–108 (2008)

Ant Based Clustering of Time Series Discrete Data – A Rough Set Approach

653

13. Pancerz, K.: Extensions of Multistage Decision Transition Systems: The Rough Set Perspective. In: Cyran, K.A., Kozielski, S., Peters, J.F., Sta´ nczyk, U., WakuliczDeja, A. (eds.) Man-Machine Interactions. AISC, vol. 59, pp. 209–216. Springer, Heidelberg (2009) 14. Pawlak, Z.: Rough Sets. Theoretical Aspects of Reasoning about Data. Kluwer Academic Publishers, Dordrecht (1991) 15. Suraj, Z.: The Synthesis Problem of Concurrent Systems Specified by Dynamic Information Systems. In: Polkowski, L., Skowron, A. (eds.) Rough Sets in Knowledge Discovery 2. Applications, Case Studies and Software Systems, pp. 418–448. Physica-Verlag, Heidelberg (1998)

Sensor Deployment for Probabilistic Target k-Coverage Using Artificial Bee Colony Algorithm S. Mini1 , Siba K. Udgata1 , and Samrat L. Sabat2 1

Department of Computer and Information Sciences University of Hyderabad, Hyderabad-500046, India [email protected], [email protected] 2 School of Physics University of Hyderabad, Hyderabad-500046, India [email protected]

Abstract. A higher level of coverage is required for many sensitive applications. Though initial work on target coverage problems in wireless sensor networks used binary sensing model, a more realistic sensing model, the probabilistic sensing model has been used later. This work considers probabilistic k-coverage; where the required level of coverage has to be satisfied with k sensors and each target should also be monitored with a specific probability. We compute the optimal deployment location of sensor nodes, such that the probabilistic coverage as well as the k-coverage requirement is satisfied with the required sensing range being optimal. Preliminary results of using artificial bee colony algorithm to solve deployment problem for probabilistic target k-coverage is reported in this paper. Keywords: Sensor Deployment, Probabilistic Coverage, Target Coverage, ABC Algorithm, k-coverage.

1

Introduction

Development of an energy-efficient wireless sensor network introduces many challenges. Coverage problem is one such issue. The widely addressed coverage problems are area coverage and target coverage. Area coverage concerns with how well a region is monitored and target coverage deals with how well point objects(targets) in the region are monitored. Since sensor nodes are battery powered, energy should be properly used to extend the network lifetime. Deployment of sensor nodes can be either random or deterministic. For a random deployment, sensor nodes can be scheduled in such a way that all nodes need not be active at the same time for coverage requirement to be satisfied. If the nodes can be deterministically deployed and if the targets are fairly large in number compared to the given number of sensor nodes, energy usage can be restricted by limiting the sensing range requirement. B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 654–661, 2011. c Springer-Verlag Berlin Heidelberg 2011 

Sensor Deployment for Probabilistic Target k-Coverage

655

Although work has been done in area coverage and target coverage problems, the results have been reported either for a binary detection model or for a probabilistic sensing model. In this paper, we address probabilistic target k-coverage for deterministic deployment of sensor nodes. The target is either monitored with full confidence or not monitored in a binary sensing model. With probabilistic model, the probability that the sensor detects a target depends on the relative position of the target within the sensors’ sensing range. Probabilistic coverage applies with some kinds of sensors e.g. acoustic, seismic etc., where the signal strength decays with distance from the source, and not with sensors that only measure local point values e.g. temperature, humidity, light etc.[1]. It can be used for applications which require a certain degree of confidence. In probabilistic k-coverage, given a region with n sensor nodes and m targets, each target needs to be monitored by at least k sensors, where 1 ≤ k ≤ n, as well as monitored with a required probability p for the network to function. We use Artificial Bee Colony algorithm [2][3][4] to compute the optimal deployment locations so that the sensing range requirement is at minimum, satisfying kcoverage and probabilistic coverage. The rest of the paper is organized as follows: Section 2 presents an overview of related work. In Section 3, the problem is defined. We present the proposed method in Section 4. The proposed method is evaluated by simulations in Section 5. Section 6 concludes the paper.

2

Related Work

Ahmed et al. [1] propose a probabilistic coverage algorithm to evaluate area coverage in a randomly deployed wireless sensor network. The coverage issues in wireless sensor networks are investigated based on probabilistic coverage and propose a distributed Probabilistic Coverage Algorithm (PCA) to evaluate the degree of confidence in detection probability provided by a randomly deployed sensor network. Zou et al. [5] address the problem of selecting a subset of nodes that are active for both sensing and communication. The active node selection procedure is aimed at providing the highest possible coverage of the sensor field, i.e., the surveillance area. It also assures network connectivity for routing and information dissemination. Carter et al. [6] address area coverage problem for probabilistic deployment of sensor nodes. The problem is formulated as an optimization problem with the objective to minimize cost while covering all points above a given probability of detection coverage threshold. A probabilistic coverage matrix is defined to decouple the coverage determination method from the model. A GA (Genetic Algorithm) approach is presented to solve the optimization problem. Hefeeda et al. [7] propose and evaluate a fully distributed, probabilistic coverage protocol. Experimental study shows that the protocol activates less sensors than the others while maintaining the same level of coverage. Udgata et al. [8] use artificial bee colony algorithm to solve sensor deployment problem in irregular terrain. This addresses area coverage problem where

656

S. Mini, S.K. Udgata, and S.L. Sabat

the area under consideration is an irregular terrain. Mini et al. [9] solve sensor deployment problem for simple coverage problem in 3-D terrain using artificial bee colony algorithm where the optimal deployment position such that the required sensing range is minimum for each target to be monitored by at least one sensor node, is computed. Mini et al. [10] use artificial bee colony algorithm to solve sensor deployment problem for simple,k and Q-coverage problems in 3-D terrain. For simple, k and Q-coverage, the optimal deployment position where the required sensing range is minimum is identified. [8][9][10] focus on solving deployment problem in wireless sensor network using binary sensing model. Unlike all the above works, in this paper, we address sensor deployment problem for probabilistic k-coverage.

3

Problem Definition

Let S = {S1 , S2 , . . . , Sn } be the set of sensor nodes and T = {T1 , T2 , . . . , Tm } be the set of targets in a given region. A sensor node located at (x1 , y1 , z1 ) can cover a target at (x2 , y2 , z2 ) if the euclidean distance between the sensor node and the target is less than or equal to the sensing range r.  (x1 − x2 )2 + (y1 − y2 )2 + (z1 − z2 )2 ≤ r (1) 3.1

Binary Sensing Model

A binary sensing model is given by,  Mij =

1 0

if dij ≤ r, otherwise

(2)

where i = 1, 2, . . . , m and j = 1, 2, . . . , n. dij corresponds to the euclidean distance between Ti and Sj 3.2

Probabilistic Coverage

As in [5], we use the following exponential function to represent the confidence level in the received sensing signal:  e−αdij if dij ≤ r, Mij = (3) 0 otherwise where 0 ≤ α ≤ 1 is a parameter representing the physical characteristics of the sensing unit and environment. The coverage of a target Ti which is monitored by multiple sensor nodes Si is given by,  Mi (Si ) = 1 − (1 − Mij ) (4) Sj ∈Si

Sensor Deployment for Probabilistic Target k-Coverage

3.3

657

k-Coverage

Given a set of targets T = {T1 , T2 , . . . , Tm } located in a region and a set of sensor nodes S = {S1 , S2 , . . . , Sn }, each target in T should be monitored by at least k number of sensor nodes, 1 ≤ k ≤ n. 3.4

Mean of Location Points

The mean value of the location points (xq , yq , zq ) for q = 1, 2, · · · N , is represented by (a1 , a2 , a3 ), where N q=1 (xq ) a1 = (5) N N q=1 (yq ) a2 = (6) N N q=1 (zq ) a3 = (7) N 3.5

Sensor Deployment to Achieve Probabilistic Target k-Coverage

Given a set of m targets T = {T1 , T2 , . . . , Tm } located in u × v × w region and n sensor nodes S = {S1 , S2 , . . . , Sn }, place the nodes such that each target is monitored by at least k-sensor nodes and with a total probability p and sensing range is minimum. The objective is to cover each target with at least k sensor nodes and probability p and to minimize the function F = ∀j ((max(distance(Sj , Hg ))))

(8)

where H is the set of all targets monitored by Sj , j = 1, 2, . . . , n , g = 1, 2, . . . , h, where h is the total number of targets Sj monitors.

4

Proposed Method

Let the solution population be B. The region is assumed to have only stationary targets. Each solution Be = {(x1 , y1 , z1 ), (x2 , y2 , z2 ), . . . , (xn , yn , zn )} where e = 1, 2, . . . , nb , nb the total number of bees and n the total number of nodes to be deployed, corresponds to a bee. The initial solution is generated in such a way that all the targets can be probabilistically covered, and no sensor node is left idle without contributing to probabilistic k-coverage. The sensor nodes which can make each target Ti meet the required probability is then identified. Let this subset be Ri . If Ri satisfies k-coverage requirement of Ti , Ti is assigned to each sensor node in Ri . If it does not satisfy k-coverage, then identify nearest nodes which do not belong to Ri that can make Ti k-covered, along with Ri . Ti is assigned to these new sensor nodes in addition to Ri . Each sensor node is then moved to the center of all the targets which are assigned to it. This move is not

658

S. Mini, S.K. Udgata, and S.L. Sabat

Algorithm 1. Proposed Scheme 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14:

Initialize the solution population B Evaluate fitness Produce new solutions based on probabilistic coverage and k-coverage Choose the fittest bee cycle = 1 repeat Search for new solutions in the neighborhood if new solution better than old solution then Memorize new solution and discard old solution end if Replace the discarded solution with a new randomly generated solution Memorize the best solution cycle = cycle + 1 until cycle = maximumcycles

allowed if some target will not be probabilistically covered due to this shift of location. The Euclidean distance between each target and the sensor location to which it is associated is used as the fitness function to evaluate the solutions. Each sensor node is associated to a cluster, where a cluster corresponds to the set of targets monitored by the sensor node. Let Dj = (Dj1 , Dj2 , Dj3 ) be the initial position of j th cluster. F (Dj ) refers to the nectar amount at food source located at Dj . After watching the waggle dance of employed bees, an onlooker goes to the region of Dj with probability Pj defined as, F (Dj ) Pj =  n l=1 F (Dl )

(9)

where n is the total number of food sources. The onlooker finds a neighborhood food source in the vicinity of Dj by using, Dj (t + 1) = Dj (t) + δji × v

(10)

where δji is the neighborhood patch size for ith dimension of j th food source, v is random uniform variate ∈ [-1, 1] and calculates the fitness value. It should be noted that the solutions are not allowed to move beyond the edge of the region. The new solutions are also evaluated by the fitness function. If any new solution is better than the existing one, choose that solution and discard the old one. Scout bees search for a random feasible solution. The solution with the least sensing range is finally chosen as the best solution. Algorithm 1. explains the proposed scheme.

5

Results and Discussion

Initially, we consider a 10 × 10 × 10 grid for experiments. The number of targets is 10 and the number of sensor nodes is 5. The number of bees is taken as 10,

Sensor Deployment for Probabilistic Target k-Coverage

659

Table 1. Sensing Range for Probabilistic Coverage Sensing Range α 0.05

0.1

0.15 0.2

Probability 0.6 0.7 0.8 0.9 0.6 0.7 0.8 0.9 0.6 0.7 0.8 0.6 0.7

Best 2.0616 2.0616 2.0616 2.0616 2.0616 2.0616 2.0616 3.8748 2.0616 2.0616 3.8586 2.0616 3.5618

Mean 2.0616 2.0616 2.0616 2.0616 2.0616 2.0616 2.0616 3.9071 2.0616 2.0616 4.0558 2.0616 3.6170

Standard Deviation 0 0 0 0 0 0 0 0.0465 0 0 0.3406 0 0.0927

number of cycles is 500, limit for neighborhood search is 20 and the number of runs is 3. MATLAB 2007a is used for implementation. 5.1

Probabilistic Coverage

Initially, we compute the sensing range required for probabilistic coverage without considering k. The required probability for coverage is varied from to 0.6 to 0.9. α is varied from 0.05 to 0.2. Table 1. shows the sensing range requirement for probabilistic coverage. When α = 0.05, the required sensing range does not change for any of the required probability. But when α increases to 0.1, the sensing range required increases for a detection probability of 0.9. Similarly, a variation in sensing range is observed at 0.8 for α = 0.15 and at 0.7 for α = 0.2. This implies that for higher α, the sensing range requirement varies at a smaller detection probability. 5.2

Probabilistic k-Coverage

To observe the difference in sensing range required for probabilistic k-coverage, k takes values 2 and 3. The required probability for coverage is varied from 0.6 to 0.9 and α is varied from 0.05 to 0.2. Table 2. shows the sensing range requirement for probabilistic k-coverage. For a constant detection probability, the sensing range requirement may or may not increase with k. This is because for some cases, more than k sensor nodes may have to monitor a target for satisfying probabilistic coverage. Due to the same reason, there are instances where probabilistic coverage and probabilistic kcoverage requires the same sensing range. For example, α = 0.05 and probability = 0.6 for probabilistic coverage requires sensing range of 2.0616 units, and α = 0.05, probability = 0.6, k = 1 for probabilistic k-coverage requires the same sensing range. But in this case, when k = 2, the sensing range required increases. We also consider a 100 × 100 × 20 grid for experimentation. Three instances of 100 targets being monitored by 10 sensor nodes are considered. k is varied from

660

S. Mini, S.K. Udgata, and S.L. Sabat

Table 2. Sensing Range for Probabilistic Target k-Coverage Sensing Range α

Probability 0.6 0.7

0.05

0.8 0.9 0.6 0.7

0.1

0.8 0.9 0.6

0.15

0.7 0.8 0.6

0.2

0.7

k 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3

Best 3.9249 6.1847 3.7814 6.1974 3.9514 6.1901 3.8407 6.1847 3.9175 6.1847 3.9098 6.2043 3.9105 6.1870 3.9206 6.1968 3.9538 6.1847 3.9236 6.1847 3.9286 6.2005 3.9319 6.1851 3.9119 6.1847

Mean 3.9844 6.2085 3.9384 6.2051 3.9802 6.2034 3.9497 6.2054 3.9870 6.2235 3.9505 6.2235 3.9433 6.2013 4.2680 6.2175 3.9658 6.2187 3.9490 6.2129 4.0131 6.2144 3.9531 6.1876 3.9789 6.1963

Standard Deviation 0.0531 0.0315 0.1550 0.0089 0.0251 0.0142 0.1330 0.0187 0.1050 0.0366 0.0423 0.0262 0.0289 0.0125 0.3030 0.0319 0.0107 0.0296 0.0304 0.0289 0.0757 0.0128 0.0198 0.0042 0.0749 0.0113

Table 3. Experimental Results Sensing Range k 1

2

3

4

5

Instance 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

Best 19.2000 19.2428 18.6335 28.3267 29.5044 28.3521 41.8409 43.9295 43.7273 49.8290 50.3353 50.7494 51.6985 52.192 54.0340

Mean 19.3030 19.3317 19.0473 28.5713 29.7045 28.5780 42.5757 44.0164 44.0340 49.8973 50.5659 51.2090 52.2129 52.498 54.4637

Standard Deviation 0.2485 0.1487 0.3584 0.2630 0.3058 0.1957 0.6741 0.1469 0.5125 0.0611 0.2514 0.7510 0.4852 0.3183 0.5567

1 to 5. The required probability is set to 0.8 and α is assumed to be 0.01. Table 3. shows the sensing range requirement for this set-up. There is no significant variation in standard deviation even with an increase in k. This shows that the method is a reliable one even for higher k.

Sensor Deployment for Probabilistic Target k-Coverage

6

661

Conclusion and Future Work

This paper explores the use of both probabilistic coverage and probabilistic kcoverage models for target coverage problem and proposes a method to compute the optimal deployment locations so that the sensing range requirement is minimum. ABC algorithm has been used to solve both probabilistic coverage and probabilistic k-coverage. The variation in sensing range is studied for a range of detection probabilites (p), coverage requirement (k) and physical medium characteristics (α). ABC algorithm proves to be reliable in getting the optimal deployment locations. The standard deviation of obtained sensing range among various runs does not change significantly for a larger region or for higher values of k. In future, we plan to extend this work for probabilistic Q-coverage.

References 1. Ahmed, N., Kanhere, S.S., Jha, S.: Probabilistic Coverage in Wireless Sensor Networks. In: Proc. of IEEE LCN 2005, pp. 672–681 (2005) 2. Karaboga, D., Basturk, B.: A powerful and efficient algorithm for numerical function optimization: artificial bee colony (ABC) algorithm. Journal of Global Optimization 39, 459–471 (2007) 3. Karaboga, D., Basturk, B.: On the performance of artificial bee colony (ABC) algorithm. Applied Soft Computing 8, 687–697 (2008) 4. Karaboga, D., Akay, B.: A survey: algorithms simulating bee swarm intelligence. Artificial Intelligence Review 31, 61–85 (2009) 5. Zou, Y., Chakrabarty, K.: A Distributed Coverage- and Connectivity-Centric Technique for Selecting Active Nodes in Wireless Sensor Networks. IEEE Transactions on Computers 54(8), 978–991 (2005) 6. Carter, B., Ragade, R.: A probabilistic model for the deployment of sensors. In: Sensors Applications Symposium, pp. 7–12 (2009) 7. Hefeeda, M., Ahmadi, H.: Energy-Efficient Protocol for Deterministic and Probabilistic Coverage in Sensor Networks. IEEE Transactions on Parallel and Distributed Systems 21(5), 579–593 (2010) 8. Udgata, S.K., Sabat, S.L., Mini, S.: Sensor Deployment in Irregular Terrain using ABC Algorithm. In: Proc. of IEEE BICA 2009, pp. 296–300 (2009) 9. Mini, S., Udgata, S.K., Sabat, S.L.: Sensor Deployment in 3-D Terrain using Artificial Bee Colony Algorithm. In: Panigrahi, B.K., Das, S., Suganthan, P.N., Dash, S.S. (eds.) SEMCCO 2010. LNCS, vol. 6466, pp. 424–431. Springer, Heidelberg (2010) 10. Mini, S., Udgata, S.K., Sabat, S.L.: Artificial Bee Colony based Sensor Deployment Algorithm for Target Coverage Problem in 3-D Terrain. In: Natarajan, R., Ojo, A. (eds.) ICDCIT 2011. LNCS, vol. 6536, pp. 313–324. Springer, Heidelberg (2011)

Extended Trail Reinforcement Strategies for Ant Colony Optimization Nikola Ivkovic1, Mirko Malekovic1, and Marin Golub2 1

Faculty of Organization and Informatics, University of Zagreb {nikola.ivkovic,mirko.malekovic}@foi.hr 2 Faculty of Electrical Engineering and Computing, University of Zagreb [email protected]

Abstract. Ant colony optimization (ACO) is a metaheuristic inspired by the foraging behavior of biological ants that was successfully applied for solving computationally hard problems. The fundamental idea that drives the ACO is the usage of pheromone trails for accumulating experience about the problem that is been solved. The best performing ACO algorithms typically use one, in some sense “the best”, solution to reinforce trail components. Two main trail reinforcement strategies are used in ACO algorithms: iteration best and global best strategy. This paper extends the reinforcement strategies by using the information from an arbitrary number of previous iterations of the algorithm. The influence of proposed strategies on algorithmic behavior is analyzed on different classes of optimization problems. The conducted experiments showed that using the proposed strategies can improve the algorithm’s performance. To compare the strategies we use the Mann–Whitney and Kruskal – Wallis statistical tests. Keywords: reinforcement strategy, pheromone trail, MAX-MIN ant system, Ant colony optimization, Swarm intelligence, combinatorial optimization, parameter settings.

1

Introduction

Ant colony optimization (ACO) [1], [2] is a class of algorithms inspired by a foraging behavior of biological ants. The colony of ants searches a surrounding area for food sources. The ants that found a food leave a pheromone trail on its way back. This way, the ants communicate indirectly with the rest of the colony by modifying a surrounding environment. The other ants then follow the pheromone trails to a food source, and leave its own trails on the way back to the colony. The shorter paths are reinforced more often, and this attracts more ants, causing an autocatalytic effect. After some time, the most of ants use the shortest path. The pheromone trails laid on the longer paths eventually evaporates. The ACO algorithms use artificial ants to construct solutions using solution components. The solution components are linked with artificial pheromone trails that affect the solution construction process. The trails encompass the collective B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 662–669, 2011. © Springer-Verlag Berlin Heidelberg 2011

Extended Trail Reinforcement Strategies for Ant Colony Optimization

663

knowledge, based on the experience of the colony about the problem. At the end of iteration the components that constitute good solutions are reinforced. The ACO metaheuristic is successfully applied on a variety of hard computational problems. The first ant based algorithm, the Ant system (AS) [3], uses all solutions constructed in the previous iteration to reinforce the trails, but the trails associated with better solutions are reinforced more than lower quality solutions. After the AS, a number of ACO algorithms with improved performance on different combinatorial problems were published, and the most of them use only one solution, in some sense the best one, for trails reinforcement. The Ant colony system (ACS) [4] uses relatively greedy strategy and reinforces the trails of the global best solution (also known as best-so-far), i.e. the best solution that was constructed from the beginning of the algorithm. One of the most successful and popular [5] ACO algorithm, MAXMIN ant system (MMAS) [6] usually uses the best solution constructed in the current iteration, but for bigger problems it is recommended to use the global best strategy. Considering that standard strategies are diametrically opposite, in this paper we propose new strategies that allow a finer control between explorativity of the iteration best and greediness of the global best strategy. The proposed strategies are not concerned with the pheromone values to be added on the trails and are also applicable when the amount of additional pheromone changes as the algorithm progresses [7]. This paper is organized as following. Section 2 briefly explains the ACO algorithm. In the section 3, the new strategies for selecting a solution for trails reinforcement procedure are introduced and compared. The section 4 briefly presents optimization problems used in the experimental evaluations of the new strategies and explains conditions under which the experiments were conducted. The section 5 presents and analyzes the results of the experimental researches, and section 6 gives final conclusions.

2

Ant Colony Optimization and MAX-MIN Ant System

Ant colony optimization can be described with the preudocode written in the table 1. In the Initialize() procedure the MMAS sets all trails to a maximum value and also the parameters of the algorithm are set. After that, the algorithm runs iteratively until satisfactory good solution is found or predefined time or number of iterations elapses. Solutions are constructed in the ConstructSolutions() procedure by adding solution components in the list of components until entire solution is constructed. The probability of selecting i-th component c(i) from the set of components Li is given for MMAS by expression (1). Parameters α and β balance between pheromone trail τc(i) and heuristic value ηc(i). Update procedure includes trails evaporation using (2) for all trails, and trails reinforcement (3) for components included in the iteration best or global best solution. In the MMAS trails are maintained within a minimum and a maximum limits. The parameter ρ is a trail evaporation rate, and the f(Sbest) gives a goodness of the solution used in the trails reinforcement procedure.

664

N. Ivkovic, M. Malekovic, and M. Golub Table 1. The ACO algorithm and the formulas used in the MMAS The ACO preudocode

Formulas used in the MMAS ·

Initialize() DO until stop conditions are met ConstructSolutions() UpdateTrails() NEXT ITERATION.

3

∑ L

1

(1)

·

·

(2) (3)

Extended Reinforcement Strategies

The first extended strategy, the κ-best strategy, is defined for any natural number κ. The best solution constructed in the previous κ iterations is used for reinforcement of the trails. Formally, the κ-best solution for iteration i can be defined (for minimization problems) with expression (4). 0

min

min ,

.

(4)

Before the algorithm reaches the iteration κ, the best solution of all constructed solutions is used for the reinforcement. For implementation of this strategy it is necessary to save up to κ solutions from the previous iterations. In the every iteration, it is necessary to save one new solution, and to delete the oldest saved solution. Therefore, it is convenient to use queue – like structure, which also allows reading all solution in the structure, when searching for the best solution. The second strategy, the κ-best strategy, is a kind of approximation of the κ-best strategy that uses solutions from at most κ previous iterations. Initially, the best solution from the first iteration is set as a κ-max-best solution. In an every following iteration, the iteration best solution is saved as the κ-max-best solution if it is better than the previously saved one or if the κ-max-best solution has not been updated for previous κ iterations. The method of selecting a κ-max-best solution is described with the preudocode: counter = counter + 1 IF ib “is better than” kappaMaxBest OR counter >= kappa DO counter = 0 kappaMaxBest = ib END IF

Initially, the counter variable is set to 0 and the best solution constructed in the first iteration is saved in the kappaMaxBest variable. The iteration best solution is saved in the ib variable. A time complexity for the κ-best strategy is O(κ), since it is necessary to search the list of up to κ elements, and κ-max-best strategy has O(1) complexity. It can be shown that κ-best and κ-max-best strategies are generalization of standard strategies. For κ-best

Extended Trail Reinforcement Strategies for Ant Colony Optimization

665

strategy, if the κ is equal or greater than the maximum iteration (MAXITER), then the solutions from all iterations are considered when searching for the best one, which is equivalent to the global best strategy. Also, for the κ-max-best strategy counter cannot exceed the MAXITER, so only a global best solution can be stored in the kappaMaxBest variable, if kappa>=MAXITER. Also, at the beginning of the algorithm, when kappa 0.5 then Xi j = pi j + β else Xi j = pi j − β · (mbest − Xi j ) · log(1/u) end if end for end for end for The objective of QPSO is to identify optimal values of (C, ε , γ ) so that the error between the predicted and target values is minimized. Thus the objective function becomes; i=k 1 minimize N = ∑ (RMSEi ) (14) i=1 k ⎧  1 n ⎪ 2 ⎪ RMSE = ⎪ n ∑i=1 [(hm )i − (hs )i ] ⎨ Cmin ≤ C ≤ Cmax subject to ⎪ εmin ≤ ε ≤ εmax ⎪ ⎪ ⎩ γmin ≤ γ ≤ γmax where hs is the simulated groundwater level value obtained from the SVM model. Thus hs depends upon the appropriate selection of the SVM parameters. In this equation, the constraints denote search ranges for the parameters C, ε and γ .

736

Ch. Sudheer et al.

4 Applications and Discussions 4.1 Description of the Study Area The study region chosen is Rentachintala, a mandal or tehsil situated in Guntur district. The monthly average rainfall and groundwater level data of the study region is available from 1985 to 2004. The groundwater level data was obtained from the Central Groundwater Board where as the average rainfall data is available from Indian Meteorological Department, Pune. The average annual normal rainfall in this region is around 681 m and the major contribution of rainfall is from the south west monsoon during the months of July-October. About 59 % of the annual precipitation falls during the south west monsoon and 26 % during the north east monsoon. Since 2001, there has been a failure of rainfall over successive years which has resulted into a decrease in the normal rainfall by 14 %. Thus, because of lack of surface water, the demand for groundwater has increased substantially, which has had a very serious impact in the depletion of the groundwater resources. Figure 1 depicts the variation of average groundwater table in the study area for the period 1985-2004. The data from November 1984 to December 2001 forms training and validation data as shown in figure 1 and the remaining data forms the testing data. It can be seen from the figure 1 that the groundwater table has decreased gradually due to an increasing demand of groundwater coupled with a decreasing trend in rainfall. Also, in the Rentachintala region there is an availability of 77 ha.m of water in the non command areas whereas the demand for ground water utilization is about 100 ha.m, hence the groundwater resources is overly exploited in the non command areas of Rentachintala Mandal. Time ( in months) Jul-83 0

Mar-86

Dec-88

Sep-91

Jun-94

Mar-97

Dec-99

Sep-02

May-05

Ground water level (m)

1

2 3 4 5

6 7 8 9

Fig. 1. Observed groundwater levels for the study period November 1984-October 2004

Groundwater Level Forecasting Using SVM-QPSO

737

4.2 Performance Measures Traditionally, the forecast performance is evaluated using the root mean square error (RMSE), Nash-Sutcliffe efficiency coefficient (EFF) and correlation coefficient (CORR). The evaluation criteria is determined by using the equation (Jain and Srinivasulu 2004): 1 n RMSE = (15) ∑ [(hm )i − (hs )i ]2 n i=1 EFF = 1 − n

∑ni=1 [(hm )i − (hs )i ]2 ∑ni=1 [(hm )i − (h¯ m )]2

[(hm )i − (h¯ m )][(hs )i − (h¯ s )]  ∑ni=1 [(hm )i − (h¯ m )]2 ∑ni=1 [(hs )i − (h¯ s )]2

CORR = ∑ i=1

(16) (17)

Where h is the groundwater level and the subscripts m and s represent the measured and simulated values respectively. The average value of associated variable is represented with a ’bar’ above it and n depicts the total number of training records. 4.3 Model Development In this study, an appropriate input data set is identified by carefully analyzing the various combinations of the groundwater levels h at various time lags. The input vector is modified each time by successively adding an groundwater level at one more time lag leading to the development of a new SVM model. As per the Central Ground Water Board (CGWB) reports the Groundwater in the region of Rentachintala is under confined conditions. The groundwater formations in this region lack primary porosity. However, it has developed secondary porosity through development of fractures and subsequent weathering over ages and has thus become water bearing. There is a possibility chance of rainwater percolating through these fractures and reaching the aquifer. This has been experimented in this study by including rainfall data (R(t))to investigate the influence of Precipitation on groundwater in this region. Further the appropriate input vector is identified by comparing the coefficient of correlation, efficiency and root mean square error obtained by performing the 5 fold cross validation test for all the models given below. Five SVM models were developed with different set of inputs variables as follows . Model 1 Model 2 Model 3 Model 4 Model 5

h(t) = h(t) = h(t) = h(t) = h(t) =

f [h(t − 1)] f [h(t − 1), h(t − 2)] f [h(t − 1), h(t − 2), h(t − 3)] f [h(t − 1), h(t − 2), h(t − 3),R(t − 1)] f [h(t − 1), h(t − 2), h(t − 3),R(t − 1), R(t − 2)]

Table 1 depicts the SVM parameters for the different models using Particle Swarm optimization technique (PSO) as mentioned in section 3 . Due to lack of any priori knowledge on the bounds of SVM parameters, a two-step PSO search algorithm [21]

738

Ch. Sudheer et al.

is recommended. In the first step, a coarse range search is made to achieve the best region of the three-dimensional grids. Since doing a complete grid-search may still be time-consuming, a coarse grid search is recommended first. After identifying a better region on the grid, a finer grid search on that region can be conducted. In the present study, the coarse range partitions for C are taken as [10−5 , 105 ]. Similarly, the coarse range partitions for ε are taken to be [0, 101 ] and the coarse range partitions for γ are [0 , 101 ]. Once the better region of grid is determined then a search is conducted in the finer range. Thus in the second stage search the parameter C ranges between [10−1 , 101 ] , ε is taken to be [10−7 , 10−1 ] and γ is taken to [0 , 1]. The program terminates when the rmse values of the SVM model is less than 10−2 or it terminates after 1000 iterations. For real field cases it is difficult to obtained such low RMSE values so in general program terminates after 1000 iterations. Table 1. Optimal SVM parameters obtained from PSO for different models

Model

C

ε

γ

Model 1 Model 2 Model 3 Model 4 Model 5

1.78 1.056 2.117 1.692 1.735

0.101 0.017 0.076 0.111 0.027

0.975 0.6 0.471 0.935 0.821

As seen from the table 2, the SVM models are evaluated based on their performance in the training and testing sets.The maximum coefficient of determination (r2 ) obtained was 0.87 (in model 2) and the lowest r2 term was 0.29 (model 4). In addition to this, model 2 exhibits the maximum value of efficiency (0.85) and minimum RMSE value (0.42). Model 2 which consists of two lags of groundwater level shows the highest efficiency, correlation and minimum RMSE. As a result, model 2 has been selected as the best-fit model to estimate the groundwater level in the Rentachintala region of the Guntur District. Table 2. Optimal SVM parameters obtained from PSO for different models

Model

r2

RMSE

Efficiency

Model 1 Model 2 Model 3 Model 4 Model 5

0.84 0.87 0.65 0.29 0.45

0.72 0.42 1.28 5.9 8.98

0.70 0.85 -0.32 1.5789 - 10.98

Groundwater Level Forecasting Using SVM-QPSO

739

5 Comparison with Other Forecasting Models The forecasting accuracy of the proposed SVM-QPSO model is compared with Neural networks. To set a direct comparison, the ANN model is trained using the same training data set (h(t) = f [h(t − 1), h(t − 2)]) as used for the SVM-QPSO. ANN model developed in this study has architecture of 2-13-1. The architecture of ANN is obtained by a trail and error procedure. Table 3 is showing the performance of comparison of SVM model with ANN. For the testing data, SVM-QPSO model has RMSE of 0.43 whereas it is 0.782 for ANFIS, 0.64 for ANN, and 1.936 for ARMA process. Thus SVM-PSO shows an improved performance compared to other models. Similarly the correlation coefficient also improves when one moves from ARMA, ANN, ANFIS to SVM-PSO. As SVM-PSO predicts the groundwater level values with a R value of 0.946 while ANFIS model exhibits R value of 0.793, ANN has R value of 0.85 and for ARMA model it is 0.769. Table 3. Comparison of SVM-PSO performance with the other forecasting models for testing data set. Performance measures

SVMQPSO

ANN

RMSE r2 Effciency

0.43 0.94 0.84

0.64 0.85 0.81

The accuracy of the lowest water level prediction in a one month time is computed as the percentage Error of the deepest water level fluctuation in the validation period (%EDLF) and is given by the equation   ho − hc %EDLF = × 100 (18) ho Where ho is the observed deepest water level in the data set and hc is the calculated water level corresponding to the observed deepest water level. Although all the models underestimate the lower groundwater levels, the SVM underestimate it by 7.5 % whereas 15 % by ANN. Observed deepest groundwater level is 9.0055 m and SVM predicts it as 9.0 m. Where as ANN predicts as 8.28 m. This shows that SVM-PSO model is able to predict the deepest groundwater level with more accuracy compared to ANN.

6 Summary and Conclusions The accurate prediction of groundwater levels is extremely helpful of planning and management of water resources. Therefore, in this study an attempt is made to develop an efficient forecasting model for predicting groundwater levels in the Rentachintala

740

Ch. Sudheer et al.

region of Andhrapradesh, India. The accuracy of SVM-QPSO model has been investigated in the present study. The SVM-QPSO model is obtained by combining two methods QPSO and SVM. SVM conducts structural minimization rather than the minimization of the errors. Further QPSO selects the optimal SVM parameters to improve the forecasting accuracy. So this unique combination of SVM and QPSO has made the proposed SVM-QPSO model to performs better compared to the other models. Also a number of performance measures like coefficient of correlation, root mean square error (RMSE) and Nash sutcliffe efficiency were considered for comparing the performance of SVM-PSO with the other forecasting techniques. In all the cases the results indicate that SVM-QPSO model forecasts are more accurate in predicting the groundwater levels. Thus SVM-QPSO model can be better alternative for forecasting groundwater level at a particular region. Further, for future work the information like use of water for agricultural and other purposes can be collected and can obtain the reasons for dips in groundwater levels at certain period of years and those can form as good input parameters.

References 1. Sivapragasam, C., Liong, S., Pasha, M.: Rainfall and runoff forecasting with SSA-SVM approach. Journal of Hydroinformatics 3, 141–152 (2001) 2. Khadam, I., Kaluarachchi, J.: Use of soft information to describe the relative uncertainty of calibration data in hydrologic models. Water Resources Research 40 (2004) W11505 3. Asefa, T., Kemblowski, M., McKee, M., Khalil, A.: Multi-time scale stream flow predictions: The support vector machines approach. Journal of Hydrology 318, 7–16 (2006) 4. Yu, X., Liong, S., Babovic, V.: Hydrologic Forecasting with Support Vector Machine Combined with Chaos-Inspired Approach. In: Hydroinformatics 2002: Proceedings of the 5th International Conference on Hydroinformatics, pp. 1–5 (2002) 5. Wang, W., Chau, K., Cheng, C., Qiu, L.: A comparison of performance of several artificial intelligence methods for forecasting monthly discharge time series. Journal of Hydrology 374, 294–306 (2009) 6. Wu, C., Chau, K., Li, Y.: River stage prediction based on a distributed support vector regression. Journal of Hydrology 358, 96–111 (2008) 7. Wu, C., Chau, K., Li, Y.: Predicting monthly streamflow using data-driven models coupled with data-preprocessing techniques. Water Resources Research 45 (2009) W08432 8. Chau, K., Wu, C.: A hybrid model coupled with singular spectrum analysis for daily rainfall prediction. Journal of Hydroinformatics 12, 458–473 (2010) 9. Lin, J., Cheng, C., Chau, K.: Using support vector machines for long-term discharge prediction. Hydrological Sciences Journal 51, 599–612 (2006) 10. Cherkassky, V., Ma, Y.: Practical selection of SVM parameters and noise estimation for SVM regression. Neural Networks 17, 113–126 (2004) 11. M¨uller, K., Smola, A., R¨atsch, G., Sch¨olkopf, B., Kohlmorgen, J., Vapnik, V.: Using support vector machines for time series prediction (2000) 12. Sch¨olkopf, B., Bartlett, P., Smola, A., Williamson, R.: Support vector regression with automatic accuracy control. In: Proceedings of ICANN 1998, Perspectives in Neural Computing, Citeseer, pp. 111–116 (1998) 13. Kennedy, J., Eberhart, R.: Particle swarm optimization. In: Proceedings of IEEE International Conference on Neural Networks, 1995, vol. 4, pp. 1942–1948. IEEE (1995)

Groundwater Level Forecasting Using SVM-QPSO

741

14. Hassan, R., Cohanim, B., De Weck, O., Venter, G.: A comparison of particle swarm optimization and the genetic algorithm. In: Proceedings of the 1st AIAA Multidisciplinary Design Optimization Specialist Conference (2005) 15. Clerc, M.: The swarm and the queen: towards a deterministic and adaptive particle swarm optimization. In: Proceedings of the 1999 Congress on Evolutionary Computation, CEC 1999, vol. 3. IEEE (1999) 16. Zheng, Y., Ma, L., Zhang, L., Qian, J.: Empirical study of particle swarm optimizer with an increasing inertia weight. In: The 2003 Congress on Evolutionary Computation, CEC 2003, vol. 1, pp. 221–226. IEEE (2003) 17. Sun, J., Xu, W., Feng, B.: A global search strategy of quantum-behaved particle swarm optimization. In: 2004 IEEE Conference on Cybernetics and Intelligent Systems, pp. 111–116. IEEE (2004) 18. Cortes, C., Vapnik, V.: Support-vector networks. Machine learning 20, 273–297 (1995) 19. Choy, K., Chan, C.: Modelling of river discharges and rainfall using radial basis function networks based on support vector regression. International Journal of Systems Science 34, 763–773 (2003) 20. Yu, X., Liong, S., Babovic, V.: EC-SVM approach for real-time hydrologic forecasting. Journal of Hydroinformatics 6, 209–223 (2004) 21. Hsu, C., Chang, C., Lin, C., et al.: A practical guide to support vector classification (2003)

Genetic Algorithm Based Optimal Design of Hydraulic Structures with Uncertainty Characterization Raj Mohan Singh Department of Civil Engineering, Motilal Nehru National Institute of Technology, Allahabad-211004, India [email protected], [email protected]

Abstract. Hydraulic structures such as weirs or barrages are integral parts of diversion head works in the alluvial plains of India. However, there is no fixed procedure to design the basic barrage parameters. The depth of sheet piles, the length and thickness of floor may be treated as basic barrage parameters. Present work discusses the procedure of optimal design using Genetic algorithm (GA). An optimization based methodology is presented to obtain the optimum structural dimensions that minimize the total cost as well as satisfy the exit gradient criteria. Nonlinear optimization formulation (NLOF) with subsurface flow embedded as constraint in the optimization formulation is solved by GA. The results obtained in this study shows that considerable cost savings can be achieved when the proposed NLOF is solved using GA. Further, uncertainty in design, and hence cost from uncertain hydrogeologic parameter, seepage head, is investigated using fuzzy numbers. Keywords: nonlinear optimization formulation, genetic algorithm, hydraulic structures, barrage design, fuzzy numbers, uncertainty characterization.

1

Introduction

A safe and optimal design of hydraulic structures is always being a challenge to water resource researchers. The seepage head causing the seepage vary with variation in flows. Design of hydraulic structures should also insure safety against seepage induced failure of the hydraulic structures. The variation in seepage head affects the downstream sheet pile depth, overall length of impervious floor, and thickness of impervious floor. The exit gradient, which is considered the most appropriate criterion to ensure safety against seepage induced piping [1, 2, 3, 4] on permeable foundations, exhibits non linear variation in floor length with variation in depth of down stream sheet pile. These facts complicate the problem and increase the non linearity of the problem. However, an optimization problem may be formulated to obtain the optimum structural dimensions that minimize the cost as well as satisfy the safe exit gradient criteria. The optimization problem for determining an optimal section for the weirs or barrages consists of minimizing the construction cost, earth work, cost of sheet piling, B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 742–749, 2011. © Springer-Verlag Berlin Heidelberg 2011

Genetic Algorithm Based Optimal Design of Hydraulic Structures

743

and length of impervious floor [5, 6]. Earlier work [5] discussed the optimal design of barrage profile for single deterministic value of seepage head. This study first solve the of nonlinear optimization formulation problem (NLOP) using genetic algorithm (GA) which gives optimal dimensions of the barrage profile that minimizes unit cost of concrete work, and earthwork and searches the barrage dimension satisfying the exit gradient criteria. The work is then extended to characterize uncertainty in design due to uncertainty in measured value of exit gradient, an important hydrogeologic parameter. Uncertainty in design, and hence cost from uncertain safe exit gradient value are quantified using fuzzy numbers. 2

Subsurface Flow

The general seepage equation under a barrage profile may be written as:

∂ 2h ∂ 2h ∂ 2h + + =0 ∂x 2 ∂y 2 ∂z 2

(1)

This is well known Laplace equation for seepage of water through porous media. This equation implicitly assumes that (i) the soil is homogeneous and isotropic; (ii) the voids are completely filled with water; (iii) no consolidation or expansion of soil takes place;and (iv) flow is steady and obeys Darcy’s law. For 2-dimensional flow, the seepage equation (1) may be written as: ∂ 2h ∂2h + =0 ∂x 2 ∂y 2

(2)

The need to provide adequate resistance to seepage flow represented by equation (1) both under and around a hydraulic structure may be an important determinant of its geometry [7]. The boundary between hydraulic structural surface and foundation soil represents a potential plane of failure. Stability under a given hydraulic head could in theory be achieved by an almost limitless combination of vertical and horizontal contact surfaces below the structure provided that the total length of the resultant seepage path were adequately long for that head [7, 8]. Present work utilized Khosla's Method of independent variables [4] to simulate the subsurface behavior in the optimization formulation. Method of independent variables is based on Schwarz-Christoffel transformation to solve the Laplace equation (1) which represents seepage through the subsurface media under a hydraulic structure. A composite structure is split up into a number of simple standard forms each of which has a known solution. The uplift pressures at key points corresponding to each elementary form are calculated on the assumption that each form exists independently. Finally, corrections are to be applied for thickness of floor, and interference effects of piles on each others.

744

R.M. Singh

An explicit check is for the stability of the hydraulic structure for soil at the exit is devised by Khosla [4]. The exit gradient for the simple profile as in Fig. 1 is given by as follows:

GE = where λ =

H 1 dd π λ

(3)

1 L [1 + 1 + α 2 ]; α = ; L is total length of the floor; and H is the see2 dd

page head.

Fig. 1. Schematic of parameters used in exit gradient

Equation (3) gives GE equal to infinity for no sheet pile at the downstream side of the floor. Therefore, it is necessary that a vertical cutoff (sheet pile) be provided at the downstream end of the floor. To prevent piping, the exit gradient is kept well below the critical values which depend upon the type of soil. The present work uses GA based optimization formulation incorporating uplift pressure and exit gradient in the optimization model to fix depth of sheet piles and length and thickness of floor. The optimization solution thus ensures safe structure with economy.

3

Optimal Design Formulation

3.1

Optimization Model

Minimize

C (L, d1, dd) = c1(f1) + c2(f2) + c3(f3) + c4(f4) + c5(f5) Subject to

SEG ≥

H dd π λ

L l ≤ L ≤L

u

(4) (5) (6)

l

u

(7)

l

u

(8)

L, d1, dd ≥ 0

(9)

d1 ≤ d1 ≤d1 dd ≤ dd ≤dd

where C (L, d1, dd) is objective function represents total cost of barrage per unit width (S/m), S is symbol for currency of INR (Indian National Rupee). The representation C is function of floor length (L), upstream sheet pile depth (d1) and downstream sheet

Genetic Algorithm Based Optimal Design of Hydraulic Structures

745

pile depth (dd); f1 is total volume of concrete in the floor per unit width for a given barrage profile and c1 is cost of concrete floor (S/m3); f2 is the depth of upstream sheet pile below the concrete floor and c2 is the cost of upstream sheet pile including driving (S/m2); f3 is the depth of downstream sheet pile below the concrete floor and c3 is the cost of downstream sheet pile including driving (S /m2); f4 is the volume of soil excavated per unit width for laying concrete floor and c4 is cost of excavation including dewatering (S /m3); f5 is the volume of soil required in filling per unit width and c5 is cost of earth filling (S /m3); SEG is safe exit gradient for a given soil formation on which the hydraulic structure is constructed and is function of downstream depth and 1 the length of the floor; λ = [1 + 1 + α 2 ]; α = dL ; L is total length of the floor; H is d 2 the seepage head ; d1 is the upstream sheet pile depth; d2 is downstream sheet pile depth; Ll, d1l, and ddl is lower bound on L, d1 and dd respectively; Lu, d1 u, dd u are upper bound on L, d1 and dd respectively. The constraint equation (5) may be written as follows after substituting the value of λ :   H L − d d {2( ) 2 − 1} 2 − 1  d d π ( SGE ) 

1/ 2

≥0

(10)

In the optimization formulation, for a give barrage profile and seepage head H, f1 is computed by estimating thickness at different key locations of the floor using Khosla’s method of independent variables and hence nonlinear function of length of floor (L), upstream sheet pile depth (d1) and downstream sheet pile depth (d2). Similarly f4, and f5 is nonlinear. The constraint represented by equation (10) is also nonlinear function of length of the floor and downstream sheet pile depth (d2). Thus both objective function and constraint are nonlinear; make the problem in the category of nonlinear optimization program (NLOP) formulation, which are inherently complex. 3.2

Characterizing Model Functional Parameters

For a given geometry of a barrage and seepage head H, the optimization model functional parameters f1, f2, f3, f4 and f5 are characterized for the barrage profile shown in Fig. 2. Intermediate sheet-piles are not effective in reducing the uplift pressures and only add to the cost of in reducing the uplift pressures and only add to the cost of the barrage [5]. In present work, no intermediate sheet piles are considered.

Fig. 2. Schematic of barrage parameters utilized in performance evaluation

746

R.M. Singh

The optimization model represented by equations (4)-(9) and the functional parameters embedded in the optimization model are solved using Genetic Algorithm on MATLAB platform. The basic steps employed in solution are available in Singh [6]. Table 1 shows physical parameters obtained by conventional methods for Fig. 2. Table 1. Barrage profile parameters

Physical parameters *L H *d1 *dd

Values (meters) 105.37 7.12 5.45 5.9

* Decision variables to be optimized 3.3

Optimization Procedure Using Genetic Algorithm

GA was originally proposed by Holland [9] and further developed by Goldberg [10]. It is based on the principles of genetics and natural selection. GA’s are applicable to a variety of optimization problems that are not well suited for standard optimization algorithms, including problems in which the objective function is discontinuous, nondifferentiable, stochastic, or highly nonlinear [11]. The GA search starts from a population of many points, rather than starting from just one point. This parallelism means that the search will not become trapped on local optima [12]. The optimization model represented by equations (4)-(10) and the functional parameters embedded in the optimization model are solved using Genetic Algorithm on MATLAB platform. The basic steps employed in solution procedure may be presented as follows: (i)

(ii) (iii) (iv) (v)

(vi) (vii) (viii) (ix)

Specification of parameters (decision variables) and hydrogeologic parameters (seepage head, and exit gradient) of problem domain in optimization formulation. Randomly generate initial population of potential values of parameters in forms of strings. Simulate seepage flow with decoded parameters to characterize f1, f2, f3, f4, and f5 to evaluate objective function satisfying constraints. Assign fitness value of each individual of population using objective function information. Stop if termination criteria satisfied, otherwise select and met the individual with high fitness value. More fit individual end up with more copies of themselves. Perform cross-over operation on the selected parent population. Perform mutation operation as in cross over operation with low probability. Obtain new population after cross-over and mutation. Go to step (iv).

Genetic Algorithm Based Optimal Design of Hydraulic Structures

747

In crossover, the offspring or children from the parents in the mating pool is determined. Mutation is performed with very low probability equal or close to the inverse of population size [13]. Such a low probability is helpful in keeping diversity in the population, and prevents the convergence of GA to local minima. The present work employed genetic algorithm code [14], and implemented it on MATLAB platform. The termination criteria is assumed to be satisfied when the population converges i.e. the average fitness of the population matches with the best fitness of the population and/or there is a little improvement in fitness with increase in number of generations.

4

Uncertainty Characterization in the Optimization Model

In the presence of limited, inaccurate or imprecise information, simulation with fuzzy numbers represents an alternative to statistical tool to handle parametric uncertainty. Fuzzy sets offer an alternate and simple way to address uncertainties even for limited exploration data sets. In the present work, the optimal design is first obtained assuming a deterministic value of hydrogelogic parameter, seepage head, in optimization model. Uncertainty in seepage head is then characterized using fuzzy numbers. The fuzzified NLOF is then solved using GA. 4.1

Uncertainty in Hydrogeologic Parameters

Uncertainty characterization of hydrogeologic parameters such as exit gradient and seepage head is based on Zadeh’s extension principle [15]. In this study only seepage head is considered to be imprecise. Input exit gradient, as imprecise parameter, is represented by fuzzy numbers. The resulting output i.e. minimum cost obtained by the optimization model is also fuzzy numbers characterized by their membership functions. The reduced TM [16] is used in the present study. The measure of uncertainty used is the ratio of the 0.1-level support to the value of which the membership function is equal to 1 [17].

5

Results and Discussions

A typical barrage profile, a spillway portion of a barrage, is chosen for illustrating the proposed approach as shown in Fig. 2. The barrage profile shown in Fig. 2 and parameters values given in Table 1 is solved employing the methodology presented in this work. During the process of optimization, the process of going into new generation continues until the fitness of the population converged i.e. average fitness of population almost matches with the best fitness. The population converged at crossover rate of 0.5, mutation rate of 0.05, the population size is 40 and the number of generations is 100. In the optimization approach the depth of sheet-piles determined from scour considerations is taken as a lower bound (3.0 m), and the upper bound is set from practical considerations and limited to 12.0 m. In present work, for performance evaluations, value of cost of concreting, c1, is taken as S 986.0/m3; cost of sheet-piling including driving, c1, is taken as S 1510.0/m2; cost of excavation and dewatering, c3, is taken as S 35.60/m3; cost of earth filling, c4, ia taken as S 11.0/m3. The optimized values of

748

R.M. Singh

parameters for a deterministic seepage head value of 7.12m and safe exit gradient equal to 1/8 are shown in Table 2. It also resulted in a smaller floor length and overall lower cost. It has shown a savings in the barrage cost ranging from 16.73 percent. The GA based approach is also compared with classical optimization approach using nonlinear constrained optimization function ‘FMINCON’ from MATLAB [18]. It can be seen from Table 3 that the reduction in cost is found to be more than six percent. Table 2. Optimized parameters for safe exit gradient equal to 1/8

Method

Conventional GA

Physical parameters Values L d1 dd 105.37 5.45 5.9 61.00 3.1 9.2

Cost ,S /m

133605.00 111250.00

Table 3. Optimized parameters for safe exit gradient equal to 1/7

Optimization Method Classical optimization GA

Physical parameters L 51.61 40.36

d1 5.45 9.16

dd 10.42 9.81

Cost, S /m

111418.00 104340.00

Earlier work discussed uncertainty characterization for fixed single range of value of uncertainty in seepage head. Earlier work [19] discussed uncertainty characterization for 15 percent uncertain seepage head. This work discusses three ranges of seepage head variations as 10 percent, 15 percent and 20 percent uncertainty in seepage head with central value of 7.12m. Each uncertain seepage head is represented in intervals by triangular fuzzy numbers. Safe exit gradient for this work is assumed to be deterministic with a fixed value of 0.175. The measure of uncertainty is estimated employing methods discussed in methodology section. The measure of uncertainty for three different uncertain seepage head scenario is found to be 16 percent, 22 percent and 31 percent respectively.

6

Conclusions

The present work also demonstrates the fuzzy based framework for uncertainty characterization in optimal cost for imprecise hydrologic parameter such as seepage head represented as interval fuzzy numbers. Left and right spread from central value of seepage head of 7.12 m is 10 percent, 15 percent and 20 percent and corresponding uncertainty in cost and hence design is 16 percent, 22 percent and 31 percent respectively. It may be concluded that uncertainty in seepage head reflects uncertain design with approximately 1.5 times more than the uncertainty in seepage head. The GA based optimization approach is equally valid for optimal design of other major hydraulic structures.

Genetic Algorithm Based Optimal Design of Hydraulic Structures

749

References 1. Khosla, A.N.: Pressure pipe observations at panjnad weir. Paper No. 160. Punjab Engineering Congress, Punjab, India (1932) 2. Khosla, A.N., Bose, N.K., Taylor, E.M.: Design of weirs on permeable foundations. CBIP Publication No. 12, Central Board of Irrigation and Power, New Delhi (1936) (reprint 1981) 3. Varshney, R.S., Gupta, S.C., Gupta, R.L.: Theory and design of irrigation structures, Nem Chand, Roorkee, vol. 2 (1988) 4. Asawa, G.L.: Irrigation and water resources engineering. New Age International, Limited Publishers, New Delhi (2005) 5. Garg, N.K., Bhagat, S.K., Asthana, B.N.: Optimal barrage design based on subsurface flow considerations. Journal of Irrigation and Drainage Engineering 128(4), 253–263 (2002) 6. Singh, R.M.: Optimal design of barrages using genetic algorithm. In: Proceedings of National Conference on Hydraulics & Water Resources (Hydro 2007), SVNIT, Surat, pp. 623–631 (2007) 7. Skutch, J.: Minor irrigation design DROP - Design manual hydraulic analysis and design of energy-dissipating structures. TDR Project R 5830, Report OD/TN 86 (1997) 8. Leliavsky, S.: Irrigation engineering: canals and barrages. Oxford and IBH, New Delhi (1979) 9. Holland, J.H.: Adaptation in natural and artificial systems. University of Michigan Press, Ann Arbor (1975) 10. Goldberg, D.E.: Genetic algorithms in search, optimization and machine learning. Kluwer Academic Publishers, Boston (1989) 11. Haestad, M., Walski, T.M., Chase, D.V., Savic, D.A., Grayman, W., Beckwith, S., Koelle, E.: Advanced water distribution modeling and management, pp. 673–677. Haestad Press, Waterbury (2003) 12. Singh, R.M.: Identification of unknown pollution sources in groundwater using artificial neural networks and genetic algorithm. Ph.D. thesis, IIT Kanpur (2004) 13. De Jong, K.A.: An analysis of the behavior of a class of genetic adaptive systems. Ph.D. dissertation, Univ. of Mich., Ann Arbor (1975) 14. Passino, K.M.: Biomimicry for optimization, control, and automation. Springer, London (2005) 15. Zadeh, L.A.: Fuzzy algorithms. Information and Control 12, 94–102 (1968) 16. Hanss, M., Willner, K.: On using fuzzy arithmetic to solve problems with uncertain model parameters. In: Proc. of the Euromech 405 Colloquium, Valenciennes, France, pp. 85–92 (1999) 17. Abebe, A.J., Guinot, V., Solomatine, D.P.: Fuzzy alpha-cut vs. Monte Carlo techniques in assessing uncertainty in model parameters. In: 4th Int. Conf. Hydroinformatics, Iowa, USA (2000) 18. Singh, R.M.: Design of barrages with genetic algorithm based embedded simulation n optimization approach. Water Resources Management 25, 409–429 (2011) 19. Singh, R.M.: Optimal hydraulic structures profiles under uncertain seepage head. In: World Renewable Energy Congress (WREC 2011), May 8-13. Linköping University, Sweden (2011)

Author Index

Abdelaziz, A.Y. I-679, II-257 Afshar, Nejat A. II-201 Agrawal, Sanjay I-159 Ahmed, Faez II-71 Al-Betar, Mohammed Azmi II-27, II-79 Alia, Osama Moh’d II-79 Anjana, A.V.R. II-267 Arulmozhiyal, R. II-310 Awadallah, Mohammed A. II-27 Babawuro, Usman I-389 Bajer, Drazen I-290 Bakshi, Tuli I-381 Balaga, Harish I-358 Banerjee, Anuradha I-520 Banerjee, Joydeep I-1 Banerjee, Tribeni Prasad II-287 Bapi, Raju S. II-166 Basia, Pooja I-618 Baskar, Subramanian I-77, I-282 Baskaran, Kaliyaperumal II-310 Benala, Tirimula Rao I-35, I-233 Bharadwaj, Kamal K. I-433 Bhargava, Mayank I-417 Bhaskar, M. Arun I-135 Bhattacharjee, Preetha I-601 Bhattacharya, Bidishna I-68 Bolaji, Asaju La’aro II-27 Bose, Sandip II-105 Chaitanya, Vedurupaka I-670 Chakraborti, Tathagata II-89 Chakraborty, Aruna I-460, I-610 Chakraborty, Niladri I-68, I-151 Chakraborty, Sumantra I-505 Chaturvedi, D.K. I-494 Chebrolu, Srilatha I-307 Chen, Li-Yueh I-248 Chittineni, Suresh II-211 Choi, Byung-Jae I-469 Chowdhury, Aritra II-105 Chowdhury, Arkabandhu I-191 Cui, Zhi-Hua II-132 Dammavalam, Srinivasa Rao Das, Apurba I-559

I-485

Das, Asit Kumar I-372 Das, Aveek Kumar I-94, I-110, I-119 Das, Swagamtam II-182 Das, Swagatam I-1, I-19, I-51, I-94, I-102, I-670, I-688, II-105, II-223, II-239, II-287 Dasgupta, Preetam I-19, I-27 Dash, Subranshu Sekhar I-85, I-135, I-167 De, Moumita II-55 Deb, Kalyanmoy I-299, II-71 Deepa, C. II-310 Deepa, S.N. I-366 Dehrouyeh, Mohammad Hadi I-407 Dehuri, Satchidananda I-35, I-233 Dehuri, S.N. II-9 Devaraj, D. I-167 Devi, B. Aruna I-366 Devi, Swapna I-127 Dhanalakshmi, Sundararajan I-282 Dheeba, J. I-349 Dhivya, Manian II-140 Dinesh, G. II-211 Dora, Lingaraj I-159 Durga Bhavani, S. II-166 Dutta, Paramartha I-520 Ekbal, Asif I-425, II-231 El-Khodary, S.M. I-679, II-257 Fazendeiro, Paulo

II-63

Garg, Ritu I-183 Gaurav, Raj II-46 Geetha, T.V. I-530 Geethanjali, M. II-267 Ghosh, Ankur I-102 Ghosh, Kuntal I-559 Ghosh, Pradipta I-1, I-199 Ghosh, Saurav II-182, II-223, II-239 Ghosh, Subhankar I-520 Girdhar, Isha I-618 Gireeshkumar, T. II-294 Golub, Marin I-662 Gupta, H.M. I-217 Gupta, Rohan I-417

752

Author Index

Halder, Anisha I-460, I-610 Halder, Udit I-19, I-27 Hanmandlu, Madasu I-217 Hannah, M. Esther I-530 Hasanuzzaman, Mohammad II-231 Hassanzadeh, Tahereh II-174 Holt, David I-452 Hong, Wei-Chiang I-248 Hui, Wu-Yin I-469 Imran, Mohammad I-539 Inoussa, Garba I-389 Islam, Sk. Minhazul II-182, II-223, II-239 Ivkovic, Nikola I-662 Jadhav, Devidas G. I-127 Jain, Amit I-626 Jain, Anuj I-399 Jain, Himanshu I-299 Jain, Nitish II-46 Jana, Nanda Dulal I-209 Janarthanan, Ramadoss I-460, I-505, I-601, I-610, II-89 Jasper, J. I-577 Javadikia, Payam I-407 Jayaprada, S. II-157 Jindal, Abhilash II-71 Kabat, Manas Ranjan II-38 Kannan, Subramanian I-77, I-282 Kannan, V. II-267 Kannapiran, B. I-341 Kant, Vibhor I-433 Khader, Ahamad Tajudin II-27, II-79 Khosa, Rakesh I-714 Konar, Amit I-460, I-505, I-601, I-610, II-89 Krishnanand, K.R. I-85, I-697 Krishna Prasad, M.H.M. I-485 Kshirsagar, Vivek II-113 Kumar, Amioy I-217, I-417 Kumar, Arun I-274 Kumar, Dirisala J. Nagendra I-315 Kumar, Gaurav II-46 Kumar, M. Jagadeesh I-135 Kumar, Piyush I-399 Kumar, Pradeep I-143 Kumar, Pravesh I-11 Kuppa, Mrithyumjaya Rao I-539

Laha, Koushik I-102 Leno, I. Jerin I-323 Lewicki, Arkadiusz I-637, I-645 Lorestani, Ali Nejat I-407 Lotfi, Shahriar I-240, II-1 Maddala, Seetha I-485 Mahadevan, Krishnan I-282 Maheswaran, Rathinasamy I-714 Mahmood, Ali Mirza I-539 Mahmoudi, Fariborz II-174 Maity, Dipankar I-19, I-27 Maji, Pradipta I-477 Majumdar, Ratul I-94, I-102, I-110 Majumder, Amit II-231 Majumder, Sibsankar I-151 Malekovic, Mirko I-662 Malik, L.G. I-265 Malik, O.P. I-494 Mallayya, Deivamani I-332 Mandal, Ankush I-119, I-199 Mandal, Kamal K. I-68, I-151 Mandal, Rajshree I-460 Mandava, Rajeswari II-79 Manikandan, R. II-191 Martinovic, Goran I-290 Mathur, Shashi I-731 Maulik, Ujjwal II-55 Meena, Yogesh Kumar II-302 Mehrotra, Kishan G. I-723 Mini, S. I-654 Mishra, Krishna K. I-274 Mitra, Anirban II-9 Mohan, Chilukuri K. I-723 Mohan, Yogeswaran II-17 Mohanta, Dusmanta K. I-706 Mohapatra, Ankita I-697 Moirangthem, Joymala I-85 Mondal, Arnab Kumar I-688 Mukherjee, Prithwijit I-119 Mukherjee, Saswati I-530 Mukhopadhyay, Anirban II-55 Murthy, J.V.R. I-315 Murthy, Pallavi I-176 Mutyalarao, M. II-122 Naderloo, Leila I-407 Naegi, Sujata I-550 Nagori, Meghana II-113 Naik, Anima II-148

Author Index Narahari Sastry, G. II-166 Nasir, Md. I-688 Nath, Hiran V. II-294 Nayak, Niranjan I-441 Osama, Reham A.

I-679, II-257

Padmanabhan, B. I-577 Padmini, S. I-176 Pagadala, Aditya I-35 Pancerz, Krzysztof I-637, I-645 Panda, Ashok Kumar II-9 Panda, Rutuparna I-159 Panda, Sidhartha I-59 Pandit, Manjaree I-585 Pandit, Nicole I-585 Panigrahi, Bijaya Ketan I-85, I-110, I-191, I-248, I-417, I-679, I-697, I-731, II-257 Pant, Millie I-11, I-593 Patel, Manoj Kumar II-38 Patel, Rahila I-265 Patra, Gyana Ranjan I-51 Patra, Moumita II-248 Patra, M.R. II-9 Pattnaik, Shyam S. I-127 Paul, Sushmita I-477 Peddi, Santhosh I-225 Perkins, Louise I-452 Perumal, Krish II-46 Phadikar, Santanu I-372 Ponnambalam, S.G. I-43, I-323, II-17 Pothiraj, Sivakumar I-569 Potluri, Anupama II-97 Potti, Subbaraj I-569 Pradeep, A.N.S. II-211 Prasad, Shitala I-399 Prasad Reddy, P.V.G.D. II-211 Prata, Paula II-63 Pullela, S.V.V.S.R. Kumar I-315 Rabbani, Hekmat I-407 Raghavi, Ch. Sudha I-233 Raghuwanshi, M.M. I-265 Raha, Souvik I-102 Rahmani, Adel T. II-201 Raj, M. Victor I-323 Rajan, C. Christober Asir I-176 Rajasekhar, Anguluri I-670 Rajesh, Vemulakonda I-539

753

Rakshit, Pratyusha I-601, I-610 Ramachandran, Baskaran I-332 Ramesh, Subramanian I-77 Ramezani, Fatemeh I-240 Rani, Manju II-302 Rao, Nalluri Madhusudana I-706 Ravindra Reddy, B. II-166 Reddy, S. Surender I-110 Roselyn, J. Preetha I-167 Rout, Pravat Kumar I-441, I-697 Routray, Sangram Kesari I-441 Roy, Anirban I-559 Roy, Diptendu Sinha I-706 Roy, Subhrajit II-182, II-223, II-239 Sabarinath, A. II-122 Sabat, Samrat L. I-654 Sadhu, Arup Kumar I-601 Saha, Nilanjan II-191 Saha, Sanchita I-425 Saha, Sriparna I-425, II-231 Salma, Umme II-278 Sanap, Shilpa A. II-113 Sanjeevi, Sriram G. I-307 Sankar, S. Saravana I-323 Sanyal, Subir kumar I-381 Sarkar, Bijan I-381 Sarkar, Soham I-51 Satapathy, Suresh Chandra I-233, I-315, II-148, II-211 Satuluri, Naganjaneyulu I-539 Selvi, S. Tamil I-349 Sengupta, Abhronil II-89 Sengupta, Soumyadip I-688 Sequeira, Pedro II-63 Shamizi, Sevin II-1 Shankar, Deepa D. II-294 Sharma, Bhuvnesh I-618 Sharma, Tarun Kumar I-593 Sharma, Vaibhav I-217 Shrivastava, Nitin Anand I-731 Si, Tapas I-209 Sil, Jaya I-209, I-372 Singh, Alok I-225, II-97 Singh, Alpna I-550 Singh, Asheesh K. I-143 Singh, Awadhesh Kumar I-183 Singh, Kumar Anurag I-626 Singh, Pramod Kumar I-626 Singh, Prashant II-319

754

Author Index

Singh, Raj Mohan I-742 Singh, V.P. I-11, I-593 Sinha, Amrita I-358 Sinharay, Arindam I-381 Sirisetti, G.S. Surya Vamsi I-35 Sivakumar, R.S. I-577 Sobha Rani, T. II-166 Soryani, Mohsen II-201 Spansel, Steven I-452 Srinivasa Rao, V. II-157 Srivastava, Praveen Ranjan I-618, II-46 Subbaraj, P. I-341 Subramani, C. I-135 Sudheer, Ch. I-731 Suganthan, Ponnuthurai Nagaratnam II-182, II-223, II-239 Sundarambal, Murugesan II-140 Tadeusiewicz, Ryszard I-637, I-645 Taneja, Monika I-618 Tapaswi, Shashikala I-585 Tiwari, Aruna I-550 Tripathi, Anshul I-585 Tripathi, Subhransu Sekhar I-59 Tripathy, Chita Ranjan II-38 Tudu, Bhimsen I-68, I-151

Udgata, Siba K. I-654, II-248 Umrao, Rahul I-494 Ungati, Jagan Mohan II-46 Vadla, Sangeetha I-618 Vaisakh, K. II-278 Vasavi, S. II-157 Verma, Gaurav I-274 Verma, Prabha I-257 Victoire, T. Aruldoss Albert I-577 Vincent, J. Oswald II-140 Vincent, Lui Wen Han I-43 Vishwakarma, D.N. I-358 Vivek, S. I-135 Vojodi, Hakimeh II-174 Xavier James Raj, M.

II-122

Yadava, R.D.S. I-257, II-319 Yang, Chun-Xia II-132 Yenduri, Sumanth I-452 Zafar, Hamim I-1, I-191, I-199 Zhao, Shizheng II-182, II-223, II-239